# Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

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## Abstract

**:**

## 1. Introduction

- (I)
- A generalization of Johansen’s proof of the Granger Representation Theorem (from MA to AR), this is Proposition 2. Consider an $I\left(1\right)$ singular vector ${\mathbf{y}}_{t}$, with dimension r, rank $q<r$, and cointegrating rank $c\ge r-q$. Assuming that $(1-L){\mathbf{y}}_{t}$ has an ARMA structure, $\mathbf{S}\left(L\right)(1-L){\mathbf{y}}_{t}=\mathbf{B}\left(L\right){\mathbf{u}}_{t}$ and that some simple additional conditions hold, ${\mathbf{y}}_{t}$ has a representation as a vector error correction mechanism (VECM) with c error correction terms:$$\mathbf{A}\left(L\right){\mathbf{y}}_{t}={\mathbf{A}}^{*}\left(L\right)(1-L){\mathbf{y}}_{t}+\mathit{\alpha}({\mathit{\beta}}^{\prime}{\mathbf{y}}_{t-1}-\mathbf{w})=\mathbf{B}\left(0\right){\mathbf{u}}_{t},$$
- (II)
- Assuming that the parameters of $\mathbf{S}\left(L\right)$ and $\mathbf{B}\left(L\right)$ may vary in an open subset of ${\mathbb{R}}^{\lambda}$, see Section 3.2 for the definition of $\lambda $, in Proposition 3 we show that all the assumptions used to obtain (4), and also the assumption that unity is the only possible zero of $\mathbf{B}\left(L\right)$, hold for generic values of the parameters. This implies that the matrices $\mathbf{A}\left(L\right)$ and ${\mathbf{A}}^{*}\left(L\right)$ are generically of finite degree, which is obviously not the case for nonsingular vectors.2

## 2. Stationary and $\mathbf{I}\left(\mathbf{1}\right)$ Singular Vectors

#### 2.1. Stationary Singular Vectors

**Definition**

**1.**

**(Zeros and Poles)**

- (i)
- a non-singular q-dimensional white-noise process ${\mathbf{u}}_{t}$,
- (ii)
- an $r\times r$ stable polynomial matrix $\mathbf{S}\left(z\right)$, with $\mathbf{S}\left(0\right)={\mathbf{I}}_{r}$,
- (iii)
- an $r\times q$ matrix $\mathbf{B}\left(z\right)$ whose rank is q for all z with the exception of a finite subset of $\mathbb{C}$, such that$${\mathbf{y}}_{t}=\mathbf{V}\left(L\right){\mathbf{u}}_{t},$$

**Remark**

**1.**

**Definition**

**2.**

**(Genericity)**Suppose that a statement Q depends on $\mathbf{p}\in \mathcal{A}$, where $\mathcal{A}$ is an open subset of ${\mathbb{R}}^{\lambda}$. We say that Q holds generically in $\mathcal{A}$, or that Q holds for generic values of $\mathbf{p}\in \mathcal{A}$, if the subset $\mathcal{N}$ of $\mathcal{A}$ where it does not hold is nowhere dense in $\mathcal{A}$, i.e., the closure of $\mathcal{N}$ in $\mathcal{A}$ has no internal points.

**Definition**

**3.**

**(Rational reduced-rank family of filters)**Assume that $r>q$ and let $\mathcal{G}$ be a set of ordered couples $\left(\mathbf{S}\right(L),\mathbf{B}(L\left)\right)$, where:

- (i)
- $\mathbf{B}\left(L\right)$ is an $r\times q$ polynomial matrix of degree ${s}_{1}\ge 0$.
- (ii)
- $\mathbf{S}\left(L\right)$ is an $r\times r$ polynomial matrix of degree ${s}_{2}\ge 0$. $\mathbf{S}\left(0\right)={\mathbf{I}}_{r}$.
- (iii)
- Denoting by $\mathbf{p}$ the vector containing the $\lambda =rq({s}_{1}+1)+{r}^{2}{s}_{2}$ coefficients of the entries of $\mathbf{B}\left(L\right)$ and $\mathbf{S}\left(L\right)$, we assume that $\mathbf{p}\in \Pi $, where Π is an open subset of ${\mathbb{R}}^{\lambda}$ such that for $\mathbf{p}\in \Pi $,(1)$\mathbf{S}\left(z\right)$ is stable,(2)$\mathrm{rank}\left(\mathbf{B}\right(z\left)\right)=q$ with the exception of a finite subset of $\mathbb{C}$.

**Proposition**

**1.**

- (I)
- Suppose that $\mathbf{V}\left(L\right)$ is an $r\times q$ matrix polynomial in L. If $\mathbf{V}\left(z\right)$ is zeroless then $\mathbf{V}\left(L\right)$ has an $r\times r$ finite-degree stable left inverse, i.e., there exists a finite-degree polynomial $r\times r$ matrix $\mathbf{W}\left(L\right)$ such that:(a)$\mathbf{W}\left(0\right)={\mathbf{I}}_{r}$,(b)$det\left(\mathbf{W}\right(z\left)\right)=0$ implies $\left|z\right|>1$,(c)$\mathbf{W}\left(L\right)\mathbf{V}\left(L\right)=\mathbf{V}\left(0\right)$. Let ${\mathbf{y}}_{t}$ be the stationary solution of $\mathbf{S}\left(L\right){\mathit{\zeta}}_{t}=\mathbf{B}\left(L\right){\mathbf{u}}_{t}$ and suppose that $\mathbf{B}\left(L\right)$ is zeroless. Then ${\mathbf{y}}_{t}$ has a finite vector autoregressive representation (VAR) $\mathbf{A}\left(L\right){\mathbf{y}}_{t}=\mathbf{B}\left(0\right){\mathbf{u}}_{t}$, where $\mathbf{A}\left(L\right)=\mathbf{N}\left(L\right)\mathbf{S}\left(L\right)$ and $\mathbf{N}\left(L\right)$ is a finite-degree left inverse of $\mathbf{B}\left(L\right)$.
- (II)
- Assume that ${\mathbf{y}}_{t}$ is the stationary solution of $\mathbf{S}\left(L\right){\mathit{\zeta}}_{t}=\mathbf{B}\left(L\right){\mathbf{u}}_{t}$, where $\left(\mathbf{S}\right(L),\mathbf{B}(L\left)\right)$ belongs to a rational reduced-rank family of filters with parameter set Π. For generic values of the parameters in Π, $\mathbf{B}\left(L\right)$ is zeroless so that ${\mathbf{y}}_{t}$ has a finite VAR representation.

#### 2.2. Fundamentalness

**Remark**

**2.**

#### 2.3. $I\left(1\right)$ Singular Vectors

**Definition**

**4.**

**(I(0), I(1) and Cointegrated vectors)**

**I(0).**An r-dimensional ARMA ${\mathbf{y}}_{t}$ with spectral density ${\mathbf{\Sigma}}_{y}\left(\theta \right)$ is $I\left(0\right)$ if ${\mathbf{\Sigma}}_{y}\left(0\right)\ne \mathbf{0}$.

**I(1).**The r-dimensional vector stochastic process ${\mathbf{y}}_{t}$ is $I\left(1\right)$ if it is a solution $(1-L){\mathit{\zeta}}_{t}={\mathbf{z}}_{t}$ where ${\mathbf{z}}_{t}$ is an r-dimensional $I\left(0\right)$ process. The rank of ${\mathbf{y}}_{t}$ is defined as the rank of ${\mathbf{z}}_{t}$.

**Cointegration.**

**Lemma**

**1.**

**Proof.**

## 3. Representation Theory for Singular $\mathbf{I}\left(\mathbf{1}\right)$ Vectors

#### 3.1. The Granger Representation Theorem (MA to AR)

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

**Assumption**

**5.**

**Assumption**

**6.**

**Remark**

**3.**

**Proposition**

**2.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

#### 3.2. Generically, $\mathbf{A}\left(L\right)$ Is a Finite-Degree Polynomial

**Definition**

**5.**

**(Rational reduced-rank family of filters with cointegrating rank c)**Assume that $r>q$, $c>0$ and $r>c\ge r-q$. Let $\mathcal{G}$ be a set of couples $\left(\mathbf{S}\right(L),\mathbf{B}(L\left)\right)$, where:

- (i)
- The matrix $\mathbf{B}\left(L\right)$ has the parameterization$$\mathbf{B}\left(L\right)=\mathit{\xi}{\mathit{\eta}}^{\prime}+(1-L){\mathbf{B}}^{*}+{(1-L)}^{2}\mathbf{E}\left(L\right),$$
- (ii)
- $\mathbf{S}\left(L\right)$ is an $r\times r$ polynomial matrix of degree ${s}_{2}\ge 0$. $\mathbf{S}\left(0\right)={\mathbf{I}}_{r}$.
- (iii)
- Denoting by $\mathbf{p}$ the vector containing the $\lambda =(r-c)(r+q)+rq({s}_{1}+2)+{r}^{2}{s}_{2}$ coefficients of the matrices $\mathbf{S}\left(L\right)$, $\mathit{\xi}$, $\mathit{\eta}$, ${\mathbf{B}}^{*}$ and $\mathbf{E}\left(L\right)$, we assume that $\mathbf{p}\in \Pi $, where Π is an open subset of ${\mathbb{R}}^{\lambda}$ such that for $\mathbf{p}\in \Pi $:(1)$\mathbf{S}\left(z\right)$ is stable,(2)$\mathrm{rank}\left(\mathbf{B}\right(z\left)\right)=q$ with the exception of a finite subset of $\mathbb{C}$,(3)$\mathrm{rank}\left(\mathbf{B}\left(1\right)\right)=\mathrm{rank}\left(\mathit{\xi}{\mathit{\eta}}^{\prime}\right)=r-c$.

**Proposition**

**3.**

**Proof.**

**Remark**

**7.**

**Remark**

**8.**

**Proposition**

**4.**

**Remark**

**9.**

#### 3.3. Permanent and Transitory Shocks

#### 3.4. VECMs and Unrestricted VARs in The Levels

- (I)
- We generate ${\mathbf{y}}_{t}$ using a specification of (14) with $r=4$, $q=3$, $d=2$, so that $c=r-q+d=3$. The $4\times 4$ matrix $\mathbf{A}\left(L\right)$ is of degree 2. The impulse-response functions are identified by assuming that the upper $3\times 3$ submatrix of $\mathbf{B}\left(0\right)$ is lower triangular (see Appendix C for details). We replicate the generation of ${\mathbf{y}}_{t}$ 1000 times for $T=100,\phantom{\rule{4pt}{0ex}}500,\phantom{\rule{4pt}{0ex}}1000,\phantom{\rule{4pt}{0ex}}5000.$
- (II)
- For each replication, we estimate a (misspecified) VAR in differences (DVAR), a VAR in the levels (LVAR) and a VECM, as in Johansen (1988, 1991), assuming known c, the degree of $\mathbf{A}\left(L\right)$ and that of ${\mathbf{A}}^{*}\left(L\right)$. For the VAR in differences the impulse-response functions for $(1-L){\mathbf{y}}_{t}$ are cumulated to obtain impulse-response function for ${\mathbf{y}}_{t}$. The root mean square error between estimated and actual impulse-response functions is computed for each replication using all 12 impulse-responses and averaged over all replications.

## 4. Cointegration of the Observable Variables in a DFM

**Proposition**

**5.**

- (i)
- ${\mathbf{x}}_{t}^{\left(p\right)}$ is cointegrated only if ${\mathit{\chi}}_{t}^{\left(p\right)}$ and ${\mathit{\u03f5}}_{t}^{\left(p\right)}$ are both cointegrated.
- (ii)
- If $p>q-d$ then ${\mathit{\chi}}_{t}^{\left(p\right)}$ is cointegrated. If $p\le q-d$ and rank$\left({\mathbf{\Lambda}}^{\left(p\right)}\right)<p$ then ${\mathit{\chi}}_{t}^{\left(p\right)}$ is cointegrated.
- (iii)
- Let ${V}^{\chi}\subseteq {\mathbb{R}}^{p}$ and ${V}^{\u03f5}\subseteq {\mathbb{R}}^{p}$ be the cointegrating spaces of ${\mathit{\chi}}_{t}^{\left(p\right)}$ and ${\mathit{\u03f5}}_{t}^{\left(p\right)}$ respectively. The vector ${\mathbf{x}}_{t}^{\left(p\right)}$ is cointegrated if and only if the intersection of ${V}^{\chi}$ and ${V}^{\u03f5}$ contains non-zero vectors. In particular, (a) if $p>q-d$ and ${c}^{\u03f5}>q-d$ then ${\mathbf{x}}^{\left(p\right)}$ is cointegrated, (b) if $p>q-d$ and ${\mathit{\u03f5}}_{t}^{\left(p\right)}$ is stationary then ${\mathbf{x}}^{\left(p\right)}$ is cointegrated.

**Proof.**

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Disclaimer

## Conflicts of Interest

## Appendix A. Proofs

#### Appendix A.1. Assumption 3 Holds Generically

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

- (i)
- Let ${\mathbf{q}}_{2}^{*}\in {\mathbf{R}}^{\nu}$ be such that none of the leading coefficients of the polynomials e, ${g}_{i}$ and ${h}_{i}$ vanishes. Of course ${\mathcal{M}}_{2,{d}_{2}}^{{\mathbf{q}}_{2}^{*}}={d}_{2}\ne 0$.
- (ii)
- Let $\stackrel{\u02c7}{z}$ be a root of ${\delta}_{2}^{{\mathbf{q}}_{2}^{*}}\left(z\right)$. If $\stackrel{\u02c7}{z}=1$ then $\stackrel{\u02c7}{z}$ is not a root of ${\delta}_{1}^{{\mathbf{q}}_{1}}\left(z\right)$ for all ${\mathbf{q}}_{1}\in {\mathbb{R}}^{\nu}$. Suppose that $\stackrel{\u02c7}{z}$ is a root of ${g}_{j}\left(z\right)$, for some j. As the parameters of the polynomials ${f}_{i}$ and ${k}_{i}$ are free to vary in ${\mathbb{R}}^{\nu}$, then, generically in ${\mathbb{R}}^{\nu}$, ${\delta}_{1}^{{\mathbf{q}}_{1}}\left(\stackrel{\u02c7}{z}\right)\ne 0$. Iterating for all roots of ${\delta}_{2}^{{\mathbf{q}}_{2}^{*}}\left(z\right)$, generically in ${\mathbb{R}}^{\nu}$, ${\delta}_{1}^{{\mathbf{q}}_{1}}\left(z\right)$ and ${\delta}_{2}^{{\mathbf{q}}_{2}^{*}}\left(z\right)$ have no roots in common. Moreover, generically in ${\mathbb{R}}^{\nu}$, ${\mathcal{M}}_{1,{d}_{1}}^{{\mathbf{q}}_{1}}={d}_{1}\ne 0$. Thus, there exists ${\mathbf{q}}_{1}^{*}$ such that (a) ${\mathcal{M}}_{1,{d}_{1}}^{{\mathbf{q}}_{1}^{*}}={d}_{1}\ne 0$, (b) ${\delta}_{1}^{{\mathbf{q}}_{1}^{*}}\left(z\right)$ and ${\delta}_{2}^{{\mathbf{q}}_{2}^{*}}\left(z\right)$ have no roots in common.
- (iii)
- Now let ${\mathbf{p}}^{*}=\left({\mathbf{q}}_{0}\phantom{\rule{4pt}{0ex}}{\mathbf{q}}_{1}^{*}\phantom{\rule{4pt}{0ex}}{\mathbf{q}}_{2}^{*}\right)$, so that$$det\left({\mathbf{M}}_{1}^{{\mathbf{p}}^{*}}\left(z\right)\right)={\delta}_{1}^{{\mathbf{q}}_{1}^{*}}\left(z\right)),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}det\left({\mathbf{M}}_{2}^{{\mathbf{p}}^{*}}\left(z\right)\right)={\delta}_{2}^{{\mathbf{q}}_{2}^{*}}\left(z\right).$$

#### Appendix A.2. if R>Q and C≤Q, Assumptions 5 and 6 Do Not Imply That e t Is a Non-Cointegrated I(0) Process.

## Appendix B. Non Uniqueness

#### Appendix B.1. Alternative Representations with Different Numbers of Error Terms

#### Appendix B.2. Uniqueness of Impulse-Response Functions

## Appendix C. Data Generating Process for the Simulations

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1 | Usually orthonormality is assumed. This is convenient but not necessary in the present paper. |

2 | To our knowledge, the present paper is the first to study cointegration and error correction representations for $I\left(1\right)$ singular vectors, the factors of $I\left(1\right)$ 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 ${x}_{it}$ and the factors ${\mathbf{F}}_{t}$, 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. |

3 | In the square case, r = q, Assumption 3 holds if and only if M(z) is unimodular. |

4 | If ${z}^{*}$ is a zero of $\mathbf{M}\left(z\right)$, multiply $\mathbf{M}\left(z\right)$ by an invertible $r\times r$ matrix ${\mathbf{Q}}_{{z}^{*}}$ such that ${z}^{*}$ is a zero of, say, the first row of ${\mathbf{Q}}_{{z}^{*}}\mathbf{M}\left(z\right)$. Then multiply by the $r\times r$ diagonal matrix with ${(z-{z}^{*})}^{-1}$ in position $(1,1)$ and unity elsewhere on the main diagonal. Iterating, all the zeros of $\mathbf{M}\left(z\right)$ are removed. |

5 | Multiplying both sides of (A3) by $(1-L)$ and using (A2), we obtain $\tilde{\mathbf{A}}\left(L\right)\mathbf{S}{\left(L\right)}^{-1}\mathbf{B}\left(L\right){\mathbf{u}}_{t}=(1-L)\tilde{\mathbf{B}}{\tilde{\mathbf{u}}}_{t}$. Comparing the spectral densities of right- and left-hand terms, it is easy to prove that ${\tilde{\mathbf{u}}}_{t}$ must be a q-dimensional, nonsingular white noise and the rank of $\tilde{\mathbf{B}}$ must be q. |

Lags | DVAR | LVAR | VECM | Lags | DVAR | LVAR | VECM | ||
---|---|---|---|---|---|---|---|---|---|

$T=100$ | 0 | 0.06 | 0.05 | 0.05 | $T=500$ | 0 | 0.02 | 0.02 | 0.02 |

4 | 0.26 | 0.18 | 0.17 | 4 | 0.23 | 0.07 | 0.07 | ||

20 | 0.30 | 0.37 | 0.22 | 20 | 0.25 | 0.14 | 0.09 | ||

40 | 0.30 | 0.45 | 0.22 | 40 | 0.25 | 0.21 | 0.09 | ||

80 | 0.30 | 0.57 | 0.22 | 80 | 0.25 | 0.32 | 0.09 | ||

$T=1000$ | 0 | 0.02 | 0.02 | 0.02 | $T=5000$ | 0 | 0.01 | 0.01 | 0.01 |

4 | 0.23 | 0.05 | 0.05 | 4 | 0.22 | 0.02 | 0.02 | ||

20 | 0.25 | 0.09 | 0.07 | 20 | 0.25 | 0.03 | 0.03 | ||

40 | 0.25 | 0.13 | 0.07 | 40 | 0.25 | 0.04 | 0.03 | ||

80 | 0.25 | 0.22 | 0.07 | 80 | 0.25 | 0.06 | 0.03 |

© 2020 by Matteo Barigozzi and Marco Lippi. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

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Barigozzi, M.; Lippi, M.; Luciani, M. Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors. *Econometrics* **2020**, *8*, 3.
https://doi.org/10.3390/econometrics8010003

**AMA Style**

Barigozzi M, Lippi M, Luciani M. Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors. *Econometrics*. 2020; 8(1):3.
https://doi.org/10.3390/econometrics8010003

**Chicago/Turabian Style**

Barigozzi, Matteo, Marco Lippi, and Matteo Luciani. 2020. "Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors" *Econometrics* 8, no. 1: 3.
https://doi.org/10.3390/econometrics8010003