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This paper establishes an extended representation theorem for unit-root VARs. A specific algebraic technique is devised to recover stationarity from the solution of the model in the form of a cointegrating transformation. Closed forms of the results of interest are derived for integrated processes up to the 4-th order. An extension to higher-order processes turns out to be within the reach on an induction argument.
keywords: Unit roots, VAR models, Cointegrated solutions,Stationarity recovering, Parallel sum.
JEL codes: C01, C02, C32
As is well known, the solution of a unit-root VAR, $\bm{A}(L)\bm{y}_{t}=\bm{\epsilon}_{t}$, crucially rests on the inversion of the matrix polynomial $\bm{A}(L)=\sum_{k=0}^{K}\bm{A}_{k}L^{k}$ and eventually of the isomorphic matrix $\bm{A}(z)$ in the complex variable $z$ about the unit root $z$=1. The solution takes the form of an integrated stochastic process, where stationarity can be recovered by a cointegrating transformation.\\ The inversion of a matrix polynomial plays a crucial role in the representation theory of linear processes. The seminal and best known contribution to this topic is the so-called Granger representation theorem by granger1981some, granger1983co and engle1987cointegration , that addressed the issue of the inversion of a matrix polynomial inherent in the MA representation of an integrated process and derived the so-called error-correction model. Since then, a stream of research and contributions have been registered on this issue, leading to a specialized literature. Among them, phillips1991optimal, phillips1990statistical, sims1990inference, stock1993simple who worked out the triangular representation, and engle1991ncointegrated, haldrup1998representations who introduced the use of the Smith-MacMillan form of $A(L)$. Johansen, johansen1985mathematical, johansen1991estimation, johansen1992representation and johansen1996likelihood, developed the Granger's representation theorem in the context of integrated VAR models and established the necessary and sufficient conditions for the occurrence of first and second-order integrated processes (see franchi2019general for a detailed discussion of the Granger representation theorem history).\\ First schumacher1991system pointed out that Johansen's $I(1)$ conditions could be restated in terms of existence of a simple pole in the inverse of the VAR autoregressive polynomial. Several authors have investigated the problem of the inversion of a matrix polynomial about a pole (see e.g., avrachenkov2001inversion, faliva2002partitioned, faliva2003new, faliva2008dynamic, faliva2011inversion, langenhop, franchi2019general) ever since. This topic has been also recently revisited from the Hilbert-space standpoint via the theory of functional time series by beare2017cointegrated, beare2019representation and franchi2019cointegration. \\ This paper develops an approach to unit-root VAR models which crucially hinges on the Laurent expansion of the inverse of the isomorphic matrix polynomial $\bm{A(z)}$. This eventually leads to determine the VAR solution, corresponding to the order of the pole of $\bm{A(z)}^{-1}$, along with its integration and cointegration properties. Closed-form expressions of the results of interest are provided for VAR models with (co)integrated solutions up to the 4-th order. By an induction argument, the analysis can be further advanced to cover processes of higher orders.
The paper develops in a twofold way. Once an extended unit-root VAR representation theorem is stated in Section 1, the paper switches to set up the required analytical apparatus, which is provided in Sections 2 and 3. Here, closed-form expressions of the principal-part matrices in the Laurent expansion of $\bm{A}(z)^{-1}$ about a unit root are worked out. The key issue of recovering stationarity, via a linear transformation of $\bm{A}(z)^{-1}$ which annihilates the principal part, is successfully faced.
The algebraic set-up of the paper pivots around the twin equalities $\bm{A}^{-1}(z)\bm{A}(z)=\textbf{I}$ and $\bm{A}(z)\bm{A}^{-1}(z)=\textbf{I}$ which hold true in a deleted neighbourhood of $z=1$. The twin equation systems that arise from the said equalities, allow to obtain informative closed-form expressions of the principal-part matrices. It can be shown that, besides the leading matrix, all the other principal-part matrices obey a regular scheme, as they can be expressed as a sum of two components: a term whose representation does not vary with the order of the pole and another which changes according to the latter. For a principal part matrix, $\bm{N}_{j}$, which weights the power $(z-1)^{-j}$, the former term turns out to be a linear combination of all the principal-part matrices, $\bm{N}_{i}$, weighting (negative) higher order powers of $(z-1)$, namely the matrices $\bm{N}_{i}$ weighting $(z-1)^{-i}$, $i=j+1,..,m$, where $m$ is the order of the pole. The coefficient matrices of the said linear combination play a crucial role in the cointegration analysis.\\ The multiplicity of the pole is determined and analyticity at $z=1$ is recovered via a transformation $\bm{P}\bm{A}^{-1}(z)$, where $\bm{P}$ is a projection operator of the principal-part on the null space. Use of the notion of parallel sum of matrices is made to find out the cointegrating relationships and their rank. When the order of the pole is multiple, the problem of annihilating the principal-part is solved by means of more operators that jointly meet the target. To this end, the role of parallel sum of matrices proves effective to combine all the projectors needed to annihilate the principal-part. Having established the results we needed, the paper turns back to the econometric side of the problem and provides in Section 4 the proof of the unit-root VAR theorem of Section 1. From an econometric standpoint, the value added by the paper is attributable to the approach to determine the unit-root VAR solutions and the innovative stationarity recovery technique. In the paper such topics are fully developed for VAR models with solutions integrated up to the $4^{th}$ order. Thus the paper crosses the virtual threshold of I(2) processes and clears the way to tackle cointegrated processes of higher order by induction. The paper is organized as follows. Section (ref) formulates the extended theorem which provides the solution of unit-root VARs together with its integration and co-integration properties. Sections (ref) and (ref) work out the analytical premise and the algebraic results, demanded to establish the main theorem of Section 1. Section 4 gives the proof of the said theorem. Section 5 provides some concluding remarks. Two appendices complete the paper, the former is devoted to parallel sums of matrices and the latter derives the results and formulas demanded by Theorem 3.1. \\
In this section an extended representation theorem is established for unit-root VAR models whose solutions are integrated processes up to the $4^{th}$ order and stationarity recovering via cointegration is thoroughly investigated. The latter is not only interesting in itself but plays a crucial role in economic analysis insofar as it offers a key to the interpretation of the long-run dynamics inherent in economic phenomena under investigation (see e.g., banerjee1993co ). As the theorem demands an $ad \enspace hoc$ analytic apparatus, its proof is postponed until the intended algebraic toolkit is made available in the newt two sections. \\ Let us now state the following
Theorem 1.1\\ Consider the VAR model:
where
Let $z$=1 be a root of multiplicity $\mu$ of the characteristic polynomial $\bm{A}(z)$, with the other roots lying outside the unit circle. Then, the solution $\bm{y}_{t}$ of (ref) is an integrated $(\bm{I})$ and co-integrated process, that is
where $m$ ($m \le \mu$) is the least positive integer for which the matrix
is non-singular. The matrices $\bm{B}_{j}$, $\bm{C}_{j}$ arise from the rank factorizations
where the subscript $\bot$ stands for orthogonal complement, and
Here $\bm{A}^{+}$ denotes the Moore-Penrose generalized inverse of $\bm{A}$,
The cointegration matrices in (ref) are
where
with
\\ Here $\bm{X: Z=X(X+Z)^{+}Z}$ denotes the parallel sum of $\bm{X}$ and $\bm{Z}$, and $\bm{G^{\top}=I-GG^{+}}$. The cointegration ranks are
with $\bm{\Gamma}\bm{\Gamma}^{+}=\bm{\Pi}_{3,4}$ and $\bm{\Xi}\bm{\Xi}^{+}=(\bm{A}^{+}\bm{A}^{[3]}\bm{C}_{0\bot}\bm{C}_{1\bot}\bm{C}_{2\bot}\bm{C}_{3\bot})^{\top})$ \\ Proof\\ Go to Section (ref)
Several facts on the inversion of a matrix polynomial about a pole must be established before proving the theorem. This is done in the following two Sections.
As the solution of a unit-root VAR model, $\bm{A}(z)\textbf{y}_{t}=\bm{\epsilon}_{t}$, crucially rests on the operator $\bm{A}^{-1}(L)$ (see e.g., faliva2008dynamic, Sections 2.3 and 2.9 ) and the algebra of matrix polynomials in the lag operator $L$ and in a complex variable $z$ are isomorphic ( dhrymes1971distributed), let us address the issue of the inversion of $\bm{A}(z)$ about a pole, $z=z_{o}$, with $z_{o}=1$ for our purposes. \\ Starting from the two basic equalities $\bm{A}^{-1}(z)\bm{A}(z)=\textbf{I}$ and $\bm{A}(z)\bm{A}^{-1}(z)=\textbf{I}$, which hold true in a deleted neighbourhood of a pole, we derive two equation systems which allow eventually to work out closed-form expressions for the coefficient matrices of the principal part of $\bm{A}^{-1}(z)$. \\ Let
be a matrix polynomial of order $n$ and degree $K$ and $z_0$ denotes a root of
Expanding $\bm{A}(z)$ about $z=z_0$ yields
As the matrix function $\bm{A}^{-1}(z)$ is analytic through the $z$-plane except for the zeros, $z_0$, of $det\bm{A}(z)$, $\bm{A}^{-1}(z)$, the following Laurent expansion
holds in a deleted neighborhood of the pole located at $z=z_0$ (see, e.g., faliva2008dynamic). Here $m$ is the order of the pole. The first term on the right-hand side of (ref) is the principal part, while $\bm{M}(z)=\sum_{j=0}^{\infty}\bm{N}_{j}(z-z_0)^{j}$ is the regular part.
In a deleted neighbourhood of $z=z_0$, the product of the right hand sides of (ref) and (ref) yields the equalities \[ \bm{I}_n=\sum_{j=-m}^{\infty}\bm{N}_j(z-z_0)^j \left(\sum_{k=0}^{K}\frac{1}{k!}\bm{A}^{(k)}(z-z_0)^k \right) =\sum_{k=-m}^{\infty}\left(\sum_{j=0}^{m+k}\frac{1}{j!}\bm{N}_{k-j}\bm{A}^{(j)}\right)(z-z_0)^k= \]
In turn, reversing the order of multiplication yields
The coefficient matrices in the right-hand side of (ref) associated with negative powers of $(z-z_0)$ are null matrices, whereas the matrix associated with $(z-z_0)^0$ is the identity matrix, that is
The same argument applies to equation (ref) and
follows accordingly. \\Hereafter, the analysis is concerned with unit roots, $z_0$=1 , which entails that $\bm{A}(1)=\bm{A}$ is a singular matrix. \\
In this section we establish under which conditions $\bm{A}^{-1}(z)$ has a pole of order $m$ at $z=z_0=1$ , derive closed form expressions of the principal-part matrices and determine linear functions of $\bm{A}^{-1}(z)$ which are analytic at $z=1$. \\ First of all, we address the issue of finding informative expressions of the principal-part coefficient matrices in $\bm{A}^{-1}(z)$. This is done in Theorem 3.1. Here, both a basic result on the leading principal-part matrix and useful representations of the non-leading ones are provided for multiple poles. These representations turn out to be the resultant of two terms: a term whose structure is maintained as the pole order changes and a term which is peculiar to the multiplicity of the pole and vanishes if the pole is simple. In order to unburden the exposition, the working out of formulas is left to an Appendix (B). \\ Afterwords, Theorem 3.2 determines the order of the pole of $\bm{A}^{-1}(z)$, gives closed-form representations of the principal-part leading matrix of the Laurent expansion of $\bm{A}^{-1}(z)$ about $z=1$ and ascertains which linear transformations recover analycity at $z=1$. \\ Theorem 3.1 \\ Let $z=z_0$=1 be a pole of $\bm{A}^{-1}(z)$. Then, the following holds \\
The first term, $\bm{\Lambda}_{\theta}$, has a structure which depends only on ${\theta}$ and plays a crucial role in determining the linear transformations which recovers stationary from the VAR solution. The matrix $\bm{\Lambda}_{\theta}$ is the sum of the discrete convolutions of $\frac{1}{j!}\bm{A}^{+}\bm{A}^{j}$ and $-\bm{N}_{-m+\theta-j}$, of -$\bm{N}_{-m+\theta-j}$ and $\frac{1}{j!}\bm{A}^{+}\bm{A}^{j}$,and of $\bm{A}^{+}\bm{A}\bm{N}_{-m+\theta-j}$, and $\frac{1}{j!}\bm{A}^{j}\bm{A}^{+}$, respectively, that is
The second term, $\bm{\Lambda}_{m,\theta}$ has a structure which depends on both $\theta$ and the order, $m$, of the pole order. In particular, the following holds
for $\theta=1$ and for some $\bm{S}_{1,m-1}$. Here $\bm{\Theta}_{1}$ and $\bm{\Theta}_{2}$ are the matrices specified in (ref) and (ref), respectively, and
for $\theta=2$ and for some $\bm{S}_{2,m-1}$. Here $\dot{\bm{A}}^{[3]}$ and $\breve{\bm{A}}^{[3]}$ denote the matrices (ref) and (ref) in Appendix B. \\Proof\\ The proof follows from the representations of the principal-part matrices $\bm{N}_{m+1}$, $\bm{N}_{m+2}$ and $\bm{N}_{m+3}$ of formulas (ref), (ref) and (ref) (Appendix B). In particular, formula (ref) rests on (ref) together with (ref) and (ref) in the said Appendix, while formula (ref) hinges on (ref) and (ref)).
At this point we can derive the results we are mostly interested in. To this end, next theorem establishes the order of the pole of $\bm{A}(z)^{-1}$ by a determinantal criterion, gives the closed-form representations of the leading matrices of the principal-part for poles of order $1\leq m\leq 4$, determines which linear forms $\bm{P}_{m}\bm{A}(z)^{-1}=\bm{P}_{m}\bm{M}(z)$ recover analyticity at $z=1$ and eventually ascertains the ranks of the matrices $\bm{P}_{m}$ which are orthogonal to the principal-part of $\bm{A}(z)^{-1}$. \\ It is worth noting that the propositions of the theorem which follows have an econometric counterpart of prominent interest, insofar as they clear the way, thanks to the isomorphism of algebras in $L$ and $z$, to the cointegration analysis in unit-root VAR models. Indeed, the order of the pole of $\bm{A}(z)^{-1}$ determines the integration order of the solution of $\bm{A}(L)\bm{y}_{t}=\boldsymbol{\epsilon}_{t}$ and the analyticity of $\bm{P}_{m}\bm{A}(z)^{-1}$ pairs off with the stationarity of the linear transformations $\bm{P}_{m}\bm{y}_{t}$, which shed light into the otherwise hidden long-run relationships of an economic system. \\ Theorem 3.2\\ Let $\bm{A}(z)$ have a possibly repeated unit root, $z$=1, and $\bm{A}=\bm{B}_0\bm{C}_0^{'}$ be a rank-factorization of the singular matrix $\bm{A}(1)=\bm{A}$. Then, the following statements hold
with $\bm{\Gamma}\bm{\Gamma}^{+}=\bm{\Pi}_{3,4}$ and $\bm{\Xi}\bm{\Xi}^{+}=(\bm{A}^{[3]}\bm{C}_{0\bot}\bm{C}_{1\bot}\bm{C}_{2\bot}\bm{C}_{3\bot})^{\top}$ \\ Proof\\ Let $m=1$, then the following
holds true because of (ref), and the other way around.\\ Pre and post-multiplication of (ref) by $\bm{C}_{0\bot}^{+}$ and $\bm{C}_{0\bot}$, respectively, and making use of (ref) leads to the equation
which is consistent if and only if $\bm{K}_{1}$ is non-singular. Solving for $\bm{Z}_{m}$ yields
and (ref) follows from (ref), accordingly.\\ About the simple pole located at $z=1$ the Laurent expansion (ref) takes the form
with $\bm{N}_{-1}$ given by (ref). \\ By inspection of (ref) it is easy to see that
It follows that $\bm{P}_{1}\bm{A}^{-1}(z)=\bm{P}_{1}\bm{M}(z)$ is analytic at $z=1$.\\ Turning back to (ref), if $\bm{K}_{1}$ is singular then the equation becomes inconsistent and we are facing a multiple pole. It follows that the right-hand sides of both (ref) and (ref) are no longer identity matrices, but null matrices instead. Eventually, the homogeneous equation
takes the place of (ref). Equation (ref) pairs off with
which follows from (34) by using the same argument. Solving the systems (ref) and (ref) for $\bm{Z}_{m}$ yields
for some $\bm{\Psi}_{m}$, with $\bm{\Phi}_{m}=\bm{C}_{1\bot}^{+}\bm{\Psi}_{m}(\bm{B}_{1\bot}^{'})^{+}$. The representation
follows from (ref), accordingly.\\ Now, let $m=2$. Then, the following
holds true because of (ref), and the other way around.\\ Pre and post-multiplying (ref) by $\bm{C}_{1\bot}^{+}\bm{C}_{0\bot}^{+} $ and $\bm{C}_{1\bot}\bm{C}_{0\bot}$, respectively, and making use of (ref) and (ref), leads to the equation
which is consistent if and only if $\bm{K}_{2}$ is non-singular. Solving for $\bm{\Phi}_{m}$ yields
and (ref) follows from (ref), accordingly. \\ About the double pole located at $z=1$ the Laurent expansion (ref) takes the form
with $\bm{N}_{-2}$ given by (ref). Then, taking into account (ref) and (ref), it is easy to see that
Upon noting that
where $\bm{\Pi}_{2}$ is given by (ref), applying Lemma A.1 in Appendix A to $\bm{P}_{1}$ and $\bm{\Pi}_{2}$ yields a projector $\bm{P}_{2}=2(\bm{P}_{1}:\bm{\Pi}_{2})$ such that $\bm{P}_{2}\bm{A}^{-1}(z)=\bm{P}_{2}\bm{M}(z)$ is analytic at $z=1$.\\ The expression (ref) follows from (ref) in Appendix A.\\ As for the rank of $\bm{P}_{2}$, formula (ref) applies yielding
as
where $\bm{\Theta}_{1}=\bm{A}^{+}\bm{A}^{(1)}\bm{C}_{0\bot}\bm{C}_{1\bot}$, $\bm{P}_{1}^{\top}=\bm{C}_{0\bot}\bm{C}_{0\bot}^{+}$, $(\bm{A}^{+})'=(\bm{B}_{0}^{+})'(\bm{C}_{0}'\bm{C}_{0})^{-1}\bm{C}_{0}'$. Since $(\bm{A}^{+})'\bm{P}_{1}^{\top}=\bm{0}$, the result $\bm{\Theta}_{1}'\bm{P}_{1}^{\top}=\bm{0}$ follows as a by-product.\\ Turning back to (ref), if $\bm{K}_{2}$ is singular, then the equation becomes inconsistent and we are facing a pole of order higher than two. It follows that the right-hand sides of (ref) and (ref) are no longer identity matrices but null matrices instead. Eventually, the homogeneous equation
takes the place of (ref). Equation (ref) pairs off with
which follows from (34) by using the same argument. Solving the systems (ref) and (ref) for $\bm{\Phi}_{m}$ yields
for some $\bm{\Psi}_{m}$, with $\bm{Z}_{m}=\bm{C}_{2\bot}^{+}\bm{\Psi}_{m}(\bm{B}_{2\bot}^{'})^{+}$ . The representation
follows from (ref), accordingly.\\ Now, let $m=3$. Then, the following
holds true because of (ref) and the other way around. \\ Pre and post-multiplying (ref) by $\bm{F}_{1}^{-}=\bm{C}_{2\bot}^{+}\bm{C}_{1\bot}^{+}\bm{C}_{0\bot}^{+}$ and $\bm{F}_{1}=\bm{C}_{0\bot}\bm{C}_{1\bot}\bm{C}_{2\bot}$, respectively, yields
as $\bm{A}\bm{F}_{1}=\bm{0}$. Now, replacing $\bm{N}_{-m+2}$, given by (ref), into (ref) gives
as $\bm{F}_{1}^{-}\bm{A}^{+}$ and $\bm{B}_{0\bot}^{'}\bm{A}^{(1)}\bm{F}_{1}$ are null matrices.\\ Then, replacing $\bm{N}_{-m+1}$, given by (ref), into (ref) gives
as $\bm{F}_{1}^{-}\bm{\Theta}_{1}$ and $\bm{B}_{1\bot}^{'}\bm{B}_{0\bot}^{'}\bm{A}^{[2]}\bm{F}_{1}$ are null matrices. \\ Equation (ref), in light of (ref) and (ref), can be also written as follows
as $\bm{F}_{1}^{-}\bm{N}_{-m}=\bm{Z}_{m}\bm{B}_{2\bot}^{'}\bm{B}_{1\bot}^{'}\bm{B}_{0\bot}^{'}$.\\ Equation (ref) is consistent if and only if $\bm{K}_{3}$ is non-singular. Solving (ref) yields
and (ref) follows, accordingly.\\ About the 3-rd order pole located at $z=1$, the Laurent expansion (ref) takes the form
with $\bm{N}_{-3}$ given by (ref). Then, taking into account formulas (ref), (ref) and (ref), it is easy to verify that
where $\bm{\Xi}_{1}=-\bm{A}^{+}\bm{A}^{(1)}\bm{N}_{-3}\bm{A}^{(1)}\bm{A}^{+}+\bm{A}^{+}\bm{A}^{(1)}\bm{C}_{0\bot}\bm{S}_{1,1}\bm{B}_{0\bot}^{'}$.\\ Upon noting that
the application of Lemma A.1 in Appendix A leads to the conclusion
where $\bm{\Pi}_{3}$ is the matrix given by (ref), and eventually that
where $\bm{P}_{3}=2(\bm{P}_{1}:\bm{\Pi}_{3})$. In light of (ref), $\bm{P}_{3}\bm{A}^{-1}(z)$ is analytic at $z=1$. The expression (ref) follows from (ref) in Appendix A.\\ The rank of $\bm{P}_{3}$ can be established following an argument similar to that used to obtain the rank of $\bm{P}_{2}$. Applying formula (ref) in Appendix A yields
as
Here $\widetilde{\bm{\Theta}}_{1}^{\top}=\bm{A}^{+}\bm{A}^{(1)}\bm{C}_{0\bot}$, $\bm{\Theta}_{2}=\bm{A}^{+}\bm{A}^{[2]}\bm{C}_{0\bot}\bm{C}_{1\bot}\bm{C}_{2\bot}$ and use has been made of (ref) in Appendix A, of (ref) above and of the equality $\bm{\Theta}_{2}^{\top}\bm{P}_{1}^{\top}=\bm{P}_{1}^{\top}$ which can be proved by using the same approach followed to obtain (ref). Thanks to (ref), formula (ref) can be worked out as follows
Turning back to (ref), if $\bm{K}_{3}$ is singular then the equation becomes inconsistent and we are facing a pole of order higher than three. It follows that the right-hand sides of (ref) and (ref) are no longer identity matrices but null matrices instead. Eventually, the homogeneous equation
takes the place of the former (ref). Equation (ref) pairs off with
which follows from (34) making use of the same argument. Solving (ref) and (ref) for $\bm{Z}_{m}$ yields
for some $\widetilde{\bm{\Psi}}_{m}$, with $\widetilde{\bm{\Phi}}_{m}=\bm{C}_{3\bot}^{+}\widetilde{\bm{\Psi}}_{m}(\bm{B}_{3\bot}^{'})^{+}$. The representation
follows from (ref), accordingly. \\ Now, let $m=4$. Then, the following
holds true because of (ref) and the other way around. \\ Pre and post-multiplying (ref) by $\bm{F}_{2}^{-}=\bm{C}_{3\bot}^{+}\bm{C}_{2\bot}^{+}\bm{C}_{1\bot}^{+}\bm{C}_{0\bot}^{+}$ and $\bm{F}_{2}=\bm{C}_{3\bot}\bm{C}_{2\bot}\bm{C}_{1\bot}\bm{C}_{0\bot}$, respectively, yields
as $\bm{A}\bm{F}_{2}=\bm{0}$. \\ By replacing $\bm{N}_{-m+3}$, given by (ref) into (ref) gives
as $\bm{F}_{2}^{-}\bm{A}^{+}$ and $\bm{B}_{0\bot}^{'}\bm{A}^{(1)}\bm{F}_{2}$ are null matrices.\\ Then, replacing $\bm{N}_{-m+2}$, given by (ref), into (ref) gives
as $\bm{F}_{2}^{-}\bm{\Theta}_{1}$ and $\bm{B}_{1\bot}^{'}\bm{B}_{0\bot}^{'}\bm{A}^{[2]}\bm{F}_{2}$ are null matrices.\\ Finally, replacing $\bm{N}_{-m+1}$, given by (ref), into (ref) gives
Equation (ref), taking into account (ref) and (ref), can be also written as
because $\bm{F}_{2}^{-}\bm{N}_{m}=\widetilde{\bm{\Phi}}_{m}\bm{B}_{3\bot}^{'}\bm{B}_{2\bot}^{'} \bm{B}_{1\bot}^{'}\bm{B}_{0\bot}^{'}$.\\ Equation (ref) is consistent if and only if $\bm{K}_{4}$ is non-singular. Solving (ref) yields
and (ref) follows, accordingly. \\ About the 4-th order pole located at $z=1$, the Laurent expansion (ref) takes the form
with $\bm{N}_{-4}$ given by (ref). \\ Taking into account (ref), (ref), (ref) and (ref), it is easy to see that
where
Upon noting that
it follows from Lemma A1 in Appendix A that
where $\bm{\Pi}_{3,4}$ is the matrix given by (ref), and
where $\bm{\Pi}_{4}$ is the matrix given by (ref). Eventually, it follows that
where $\bm{P}_{4}=2(\bm{P}_{1}:\bm{\Pi}_{4})$. In light of (ref), $\bm{P}_{4}\bm{A}^{-1}(z)$ is analytic at $z=1$. The expression (ref) follows from (ref) in Appendix A. \\ The rank of $\bm{P}_{4}$ can be established following an argument similar to that used for $\bm{P}_{3}$. Applying formula (ref) in Appendix A yields
as it can be proved that $\bm{\Pi}_{4}\bm{P}_{1}^{\top}=\bm{P}_{1}^{\top}$ by repeating the argument of formula (ref). Applying (ref) to $\bm{\Pi}_{4}$ yields
Here
in light of (ref) in Appendix A, and
by setting $\bm{\Pi}_{3,4}=\bm{\Gamma}\bm{\Gamma}^{+}$ and $(\bm{A}^{+}\bm{A}^{[3]}\bm{C}_{0\bot}\bm{C}_{1\bot}\bm{C}_{2\bot}\bm{C}_{3\bot})^{\top}=\bm{\Xi}\bm{\Xi}^{+}$.\\ Thanks to (ref), (ref) and (ref), formula (ref) can be worked out as follows
This proves (ref).
Thanks to the analytic toolkit we have settled in the previous two sections, we are eventually ready to give the following \\ Proof of Theorem 1.1 \\The VAR Model in (ref) is a linear non-homogeneous difference-equation system in matrix form whose solution can be formally written as
where the first term is a particular solution of the non-homogeneous equation and the second is the so called complementary solution. Both depend on the operator $\bm{A}^{-1}(L)$, and eventually on the matrix $\bm{A}^{-1}(z)$ as the algebras of the polynomial functions of the lag operator $L$ and of the complex variable $z$ are isomorphic (see, e.g., dhrymes1971distributed). The particular solution $\bm{A}^{-1}(L)\bm{\varepsilon}_{t}$ is composed of a (coloured) noise term and random walks up to the $m$-th order, where $m$ is the order of the pole of $\bm{A}^{-1}(z)$ at $z=1$, while the complementary solution is a polynomial in t of ($m$-1)-th degree: altogether they lead to an $m$-th order integrated process (see, e.g., faliva2008dynamic Sections 1.8 and 2.3). As for the pole order ($m$), this is established by Theorem 3.2, and (ref) ensues accordingly. \\ As for (ref)-(ref), the formulas are proved in Theorem 3.2 as well. \\ The cointegration relationship (ref) recovers stationary by annihilating the principal-part of $\bm{A}^{-1}(z)$. The matrices $\bm{P}_{m}$, for $m$=1, 2, 3, 4 as per (ref)-(ref), are obtained in the aforesaid theorem, too. The cointegration ranks in formulas (ref)- (ref) are derived in the same theorem.
The paper investigates unit-root VARs whose solutions are (co)-integrated processes up to 4-th order. This is in itself worthy of note when compared the extant literature which is almost entirely dedicated to first and second order processes. What is more, the algebraic apparatus set forth in the paper can be successfully applied to higher order processes by induction, thus paving the way to a wide-spread field of applications. It is worth noticing that the key issue of stationary recovering via cointegration is settled for integrated processes of increasing order via a cunning algebraic argument which hinges on the notion of parallel sum. \\