Cointegration in functional autoregressive processes

Cointegration in functional autoregressive processes

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Introduction

The theory of time series that take values in infinite dimensional separable Hilbert spaces, or $\mathcal{H}$-valued processes, is receiving increasing attention in econometrics. $\mathcal{H}$-valued processes allow to represent directly the dynamics of infinite-dimensional objects, such as Lebesgue square-integrable functions on a compact domain. In this way, they allow greater modeling generality with respect to models for conditional means and variances, see e.g. HK:12.

One notable special case is given by $\mathcal{H}$-valued processes $h=\psi (f)$, where $f$ a generic probability density function (pdf) and $\psi$ is an invertible transformation; the transformation is needed because the space of pdfs is convex but not linear, see petersen2016. Modeling dynamics of a whole pdf appears of interest e.g. for the income distribution, see e.g. WB:05, Piketty:14 and CKP:16.

An important early contribution to the theory of functional time series is Bos:00, where a theoretical treatment of linear processes in Banach and Hilbert spaces is developed. There, emphasis is given to the derivations of laws of large numbers and central limit theorems that allow to discuss estimation and inference for $\mathcal{H}$-valued stationary autoregressive (AR) models.

Economic applications of functional time series analysis include studies on the term structure of interest rates, see KO:08, and intraday volatility, see HHR:13 and GHK:13; additional applications can be found in the recent monographs HK:12 and KR:17 and in the review article HK:12b.

Recently CKP:16 applied Functional Principal Components Analysis (FPCA) directly on the space of densities for individual earnings and intra-month distributions of stock returns.\footnote{Bea:17 pointed out the issue that the space of density is not linear, see also BS:18b.} They found evidence of unit root persistence in a handful of coordinates of these cross-sectional distributions. The framework proposed by CKP:16 has (by construction) a finite number of $I(1)$ stochastic trends and an infinite dimensional cointegrating space. The theory is developed starting from the infinite moving average representation of the first differences of the process and the potential unit roots are identified and tested through FPCA.

Representation of $\mathcal{H}$-valued AR processes with unit roots has been recently considered in the literature. HP:WP consider $\mathcal{H}$-valued AR$(1)$ processes with compact operator and prove that an extension of the Granger-Johansen Representation Theorem, see Theorem 4.2 in Joh:96, holds in the $I(1)$ case. The corresponding common trends representation, or functional Beveridge-Nelson decomposition, displays a finite number of $I(1)$ stochastic trends and an infinite dimensional cointegrating space. They further propose an estimator for the functional autoregressive operator which builds on the results in CKP:16.

BSS:17 consider $\mathcal{H}$-valued AR$(k)$, $k\geq 1$, with compact operators if $k>1$ and no compactness assumption if $k=1$, and show that the Granger-Johansen Representation Theorem holds in the $I(1)$ case. If $k>1$, the number of $I(1)$ stochastic trends is finite and the dimension of the cointegrating space\ is infinite, while if $k=1$ this is not necessarily the case. In order to obtain the common trends representation of $\mathcal{H}$-valued AR$(k)$, $k\geq 1$, with compact operators. BS:18 are the first to employ a theorem on the inversion of analytic operator functions in GGK:90:1.\footnote{The same theorem is used here to discuss the existence of a common trends representation in Section $\ref{sec_ars}$.} They also present results on the $I(2)$ case that show that the number of $I(2)$ stochastic trends is finite and the dimension of the cointegrating space\ is infinite.

Finally, CHP:WP consider an error correction form with compact error correction operator and show that in this case the number of $I(1)$ stochastic trends is infinite and the dimension of the cointegrating space\ is finite. Moreover, they show that Granger-Johansen Representation Theorem continues to hold.

This paper considers a more general class of AR processes, called the class of ARs with a unit root of finite type. This class contains $\mathcal{H}$-valued ARs with compact operators as a special case. This paper derives a generalization of the Granger-Johansen Representation Theorem for this class, valid for any integration order $d=1,2,\dots$ .

An existence theorem is provided; this shows that the solution of an AR with a unit root of finite type\ is necessarily $I(d)$ for some finite integer $d$ and displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks and an infinite dimensional cointegrating space. This result is a direct consequence of a well known theorem in operator theory, and first employed in BS:18 in the context of $\mathcal{H}$-valued ARs with compact operators.

Despite these interesting implications, this existence result does not address a number of important issues, such as the connections between the structure of the AR operators and $(i)$ the order of integration of the process, $(ii)$ the structure of the attractor space\ and the cointegrating space\ and $(iii)$ the expression of the cointegrating relations. The characterization of these links in the generic $I(d)$ case constitutes the main contribution of the present paper. More specifically, a necessary and sufficient condition for the order of integration $d$ is given in terms of the decomposition of the space $\mathcal{H}$ into the direct sum of $d+1$ orthogonal subspaces $\tau_h$, $h=0,\dots,d$, that are expressed recursively in terms of the AR operators. This condition is called the `pole{$(d)$} condition', because it is a necessary and sufficient condition for the inverse of the $A(z)$ function to have a pole of order $d$ at $z=1$.

A crucial feature of the present pole{$(d)$} conditions is that the subspaces in the orthogonal direct sum decomposition $\mathcal{H}=\tau_0 \oplus \tau_1 \oplus \cdots \oplus \tau_d$, $\tau_d\neq \{ 0 \}$, identify the directions in which the properties of the process differ. Specifically, for any nonzero $v\in \tau_0$, which is infinite dimensional, one can combine $\langle v,x_t\rangle$ with differences $\Delta^n x_{t}$ for $n=1,\dots,d-1$ to find $I(0)$ polynomial cointegrating relations. For any nonzero $v\in \tau_1$, with dimension $0 \leq \dim \tau_1 < \infty $, one can combine $\langle v,x_t\rangle$ with differences $\Delta^n x_{t}$ for $n=1,\dots,d-2$ and find $I(1)$ polynomial cointegrating relations.

This kind of feature is valid for $\tau_{2}, \dots, \tau_{d-2}$; for $\tau_{d-1}$, with $0 \leq \dim \tau_{d-1} < \infty $, one has $\langle v,x_t\rangle \sim I(d-1)$ for any nonzero $v$ in $\mathcal{H}$, without polynomial cointegration. Finally for nonzero $v$ in $\tau_{d}$, with with $0 < \dim \tau_{d} < \infty $ one has $\langle v,x_t\rangle \sim I(d)$ for any nonzero $v$, i.e. all $v$-characteristics have no cointegration. These results parallel the ones in the Triangular Representation in the finite dimensional case $\mathcal{H}=\mathbb{R}^p$ discussed in Phi:91 and SW:93; see also FP_idcoeff.

These results show that conditions and properties of ARs with a unit root of finite type\ extend those that apply in the usual finite dimensional VAR case; in particular for $\mathcal{H}=\mathbb{R}^p$ one finds the $I(1)$ and $I(2)$ results in Joh:96, and for the generic $I(d)$ case, one finds the results in FP_idcoeff. Except for the fact that the number of $I(0)$ cointegrating relations is infinite, the infinite dimensionality of $\mathcal{H}$ does not introduce additional elements in the representation analysis of ARs with a unit root of finite type.

The present results are based on orthogonal decomposition of the embedding Hilbert space. Orthogonal and non-orthogonal projections are well known concepts in econometrics. Students are usually introduced to these concepts when learning OLS and GLS, where the choice between the two is usually discussed in terms of estimation efficiency; see Phi:91b for how these arguments are modified for spectral GLS regressions methods in a cointegration context. In the context of the representation theory considered here, results can be obtained using either orthogonal or non-orthogonal projections. The present choice of orthogonal projections is found to ease exposition and to simplify the characterization of the cointegrating $v$-characteristics of the process.

The rest of the paper is organized as follows: Section $\ref{sec_setup}$ presents basic definitions and concepts, Section $\ref{sec_ars}$ discussed the assumption of unit root of finite type and reports initial existence results for a pole of finite order; Section $\ref{sec_char_1_2}$ provides a characterization of $I(1)$ and $I(2)$ ARs with a unit root of finite type\ and Section $\ref{sec_char_d}$ extends the analysis to the general $I(d)$, $d=1,2,\dots$, case. Section $\ref{sec_CONC}$ concludes.

Three Appendices collect background definitions, novel inversion results and proofs of the statements in the paper. Specifically, Appendix $\ref{app_dot}$ reviews notions on operators acting on a separable Hilbert space $\mathcal{H}$ and on $\mathcal{H}$-valued random variables; Appendix $\ref{app_lemmas}$ presents novel results on the inversion of a meromorphic operator function and Appendix $\ref{app_text}$ reports proofs of the results in the paper.

$\mathcal{H}$-valued linear process, order of integration and cointegration

This section introduces the notions of weakly stationary, white noise, linear, integrated, and cointegrated processes that take values in a separable Hilbert space $\mathcal{H}$, where separable means that $\mathcal{H}$ admits a countable orthonormal basis. Basic definitions of operators acting on $\mathcal{H}$ and of $\mathcal{H}$-valued random variables are reported in Appendix $\ref{app_dot}$.

Definitions

The definitions of weakly stationary and white noise process are taken from Bos:00, while those of linear, integrated and cointegrated process are adapted from Joh:96; they are similar to those employed in CKP:16,BSS:17,BS:18. The definition of expectation $\operatorname{E}(\cdot)$, covariance operator and cross-covariance function used in the following are reported in Appendix $\ref{app_random}$.

The notion of $\mathcal{H}$-valued white noise is introduced next.

Note that by definition any strong white noise is white noise, and any white noise process is weakly stationary. The same property holds for linear combinations of lags of a white noise process with suitable weights; this leads to the class of linear processes, introduced in Definition $\ref{def_LP}$ below.

In the definition below, the following notation is employed: $D(z_0,\rho)$ denotes the open disc $\{z\in \mathbb{C}:|z-z_0|<\rho\}$ with center $z_0\in\mathbb{C}$ and radius $0<\rho\in \mathbb{R}$ and $\mathcal{L}_\mathcal{H}$ indicates the set of bounded linear operators on $\mathcal{H}$ with norm $\| A \|_{\mathcal{L}_\mathcal{H}}=\sup_{\| v \|=1}\| Av \|$; an operator function $B(z)=\sum_{n=0}^{\infty}B_{n}(z-z_0)^n$, where $B_{n} \in \mathcal{L}_\mathcal{H}$, is said to be absolutely convergent on $D(z_0,\rho)$ if $\sum_{n=0}^{\infty}\|B_{n}\|_{\mathcal{L}_\mathcal{H}}|z-z_0|^n<\infty$ for all $z\in D(z_0,\rho)$.\footnote{Note that $\sum_{n=0}^{\infty}\|B_{n}\|_{\mathcal{L}_\mathcal{H}}|z-z_0|^n<\infty$ for all $z\in D(z_0,\rho)$ implies that $\sum_{n=0}^{\infty}B_{n}(z-z_0)^n$ converges in the operator norm to $B(z)\in\mathcal{L}_\mathcal{H}$ for all $z\in D(z_0,\rho)$, i.e. $\|\sum_{n=0}^{N}B_{n}(z-z_0)^n-B(z) \|_{\mathcal{L}_\mathcal{H}}\rightarrow 0$ as $N\rightarrow \infty$.} The lag operator is denoted by $L$ and $\Delta = 1-L$ is the difference operator.

As discussed in Section 7.1 in Bos:00, existence and weak stationarity of $u_t-\mu_{t}=\sum_{n=0}^{\infty}B_{n}\varepsilon_{t-n}$ are guaranteed by the square summability condition $\sum_{n=0}^{\infty}\|B_{n}\|_{\mathcal{L}_\mathcal{H}}^2<\infty$. Observe that the requirement that $B(z)$ is absolutely convergent on $D(0,\rho)$ for some $\rho>1$ is stronger. In fact, $\sum_{n=0}^{\infty}\|B_{n}\|_{\mathcal{L}_\mathcal{H}}|z|^n<\infty$ for all $z\in D(0,\rho)$, $\rho>1$, implies $\sum_{n=0}^{\infty}\|B_{n}\|_{\mathcal{L}_\mathcal{H}}<\infty$ and hence $\sum_{n=0}^{\infty}\|B_{n}\|_{\mathcal{L}_\mathcal{H}}^2<\infty$. This shows that $u_t-\mu_{t}$ in Definition $\ref{def_LP}$ is well defined and weakly stationary.

Moreover, $B(z)\in \mathcal{L}_\mathcal{H}$ for all $z\in D(0,\rho)$, $\rho>1$, implies that $B(1)$ is a bounded linear operator. Finally note that $B(z)$ is infinitely differentiable on $D(0,\rho)$, $\rho>1$, and the series obtained by termwise $k$ times differentiation, $\sum_{n=k}^{\infty}n(n-1)\cdots (n-k+1)B_{n}z^{n-k}$, is absolutely convergent and coincides with the $k$-th derivative of $B(z)$ for each $z\in D(0,\rho)$. Hence $\sum_{n=k}^{\infty}n(n-1)\cdots (n-k+1)\|B_{n}\|_{\mathcal{L}_\mathcal{H}}<\infty$, which for $k=1$ reads $\sum_{n=1}^{\infty}n\|B_{n}\|_{\mathcal{L}_\mathcal{H}}<\infty$; this condition is employed in CKP:16.

The notions of integration and integral operator are introduced next.

This definition coincides with Definition 3.3 in Joh:96 of an $I(d)$ process for the special case $\mathcal{H}=\mathbb{R}^p$.

Observe that a white noise process is $I(0)$ and that an $I(0)$ process is weakly stationary. In order to see that a weakly stationary is not necessarily $I(0)$, take for instance $u_t=\varepsilon_{t}-\varepsilon_{t-1}$; this process is weakly stationary, with $B(1)=0$ and hence it does not satisfy the definition of an $I(0)$ process, showing that the two concepts do not coincide. The distinction between weak stationarity and $I(0)$-ness is relevant for the definition of order of integration: in fact, the cumulation of an $I(0)$ process is necessarily $I(1)$ while the cumulation of stationary process is not necessarily so.

Following HP:WP, one can define the $v$-characteristic of $x_t$ as the scalar process $\langle v, x_t \rangle$, for any $v \in \mathcal{H}$. From Definition $\ref{def_INT}$, one can see that a generic $v$-characteristic of $x_t \sim I(d)$ is itself at most integrated of order $d$; the case when a $v$-characteristic of $x_t \sim I(d)$ is integrated of lower order $b<d$ is associated with the notion of cointegration.

As in the usual finite dimensional case, $z_t$ is cointegrated if there exists a nonzero linear combination $v$ of $z_t$ (i.e. a $v$-characteristic of $z_t$) that has lower order of integration than the original process. Observe that the attractor space\ (respectively the cointegrating space) contains $0\in \mathcal{H}$ and all nonzero $v\in \mathcal{H}$ that correspond to a $v$-characteristic of $z_t$ with the same (respectively lower) order of integration of the original process $z_t$. The null vector $0\in \mathcal{H}$ is added so as to make the cointegrating space\ a vector spaces.

The cases that have been studied in the literature correspond to finite dimensions either for the attractor or for the cointegrating space. When both of them have finite dimension, $\mathcal{H}$ is finite dimensional, so that the standard results in the literature apply. The case in which the attractor space\ is infinite dimensional and the cointegrating space\ is finite dimensional corresponds to a process with an infinite number of $I(d)$ stochastic trends and a finite dimensional cointegrating space. For $d=1$, this case has been discussed in CHP:WP and in BSS:17 for $k=1$, see Proposition $\ref{prop_BSS_k_1}$ below.

Most of the contributions in the literature have studied instead the case of an attractor space\ of finite dimension and a cointegrating space\ of infinite dimension, i.e. the case where the process has a finite number of $I(d)$ $v$-characteristics and an infinite number of $I(b)$ $v$-characteristics with $b<d$. This is the setup studied in CKP:16, HP:WP and BSS:17 for $d=1$ and in BS:18 for $d=1$ and $d=2$. This is the setting considered in the present paper as well, and it is motivated also by the next example.

Yield curve example

As an example of a Hilbert space of economic interest, consider the yield curve $x_{\circ,t}(s)$, where $s$ denotes maturity and $t$ time. In this section $t$ is omitted, unless needed for clarity.

Let $\mathcal{H}$ be the set of Lebesgue measurable functions $x_{\circ}(s)$ such that $\int_{0}^{s_{\max}}x_{\circ}^2(s)\mathrm{d} s<\infty $, where $s_{\max}$ is the maximal maturity. One can rescale the maturity $s$ into $u=s/s_{\max}$ and define the rescaled yield curve $x(u)$ by $x(u)=x_{\circ}(u \cdot s_{\max})$, with $u\in (0,1]$ and $\int_{0}^{1}x^{2}(u)\mathrm{d} u<\infty $. The vector space operations on $\mathcal{H}$ are defined in a natural way as $(x+y) (u) =x(u) +y(u) $ and $\left( \alpha x\right) (u) =\alpha x(u) $ where $\alpha \in \mathbb{R}$. Next define the inner product

This space of Lebesgue square-integrable functions equipped with the inner product (ref) is a complete, separable Hilbert space, see e.g. KR:17.

The yield curve is often described in terms of the three features of level, slope and curvature, see e.g. CP:05. These features of the yield curve can be associated with the following $v$-characteristics of $x$. Define $\pi _{j,1},\dots ,\pi _{j,j}$ as a partition of the unit interval $(0,1]$ into $j$ segments $\pi _{j,i}$ of length $1/j$, $\pi _{j,i}=(\frac{i-1}{j},\frac{i}{j}]$, and let $1_{\{u\in \pi _{j,i}\}}$ be the indicator function that takes value one when $u\in \pi _{j,i}$ and equals 0 otherwise.

Next define the following $v$ functions

and observe that they belong to $\mathcal{H}$, because they are Lebesgue square-integrable functions. Finally let $x$ denote the rescaled yield curve and note that

One can see that $\langle v_{0},x\rangle$ computes the average yield curve, and hence can be associated with the level of the yield curve. Similarly $\langle v_{1},x\rangle$ computes the difference between the average yield on the longer maturities and the one on the shorter maturities;\ hence it can be associated with the slope of the yield curve. Finally $\langle v_{2},x\rangle$ computes the difference of the slopes on the longer maturities and the shorter maturities; hence it can be associated with the curvature of the yield curve.

This shows that $v_{0}$, $v_{1}$, $v_{2}$ define interesting $v$-characteristics for the yield curve $x$. If the yield curve $x$ is modeled as a functional time series, $x_{t}$, then it is interesting to ask questions of the type: “what is the order of integration of the level (or slope, or curvature) of the yield curve?”. These questions translate into “what is the order of integration of the $v_{j}$-characteristics, $j=0,1,2$, of the yield curve $x_{t}$?”.

This illustrates how interesting hypotheses can be formulated in this context; clearly other types of hypotheses can be formulated in a similar way. Moreover, it is of interest to determine how many and which characteristics are nonstationary, which corresponds to estimating the (dimension of the) attractor space.

It appears natural in this context to assume (or test) that there are only a finite number of factors driving the dynamics of the yield curve. This translated into the hypothesis that there are only a finite number of nonstationary $v$-characteristics; in this case, $x_t$ would have a finite dimensional attractor space\ and an infinite dimensional cointegrating space. This seems to be a reasonable assumption to be tested empirically also beyond the case of the yield curve; this case is the one studied in the present paper, see Corollary $\ref{coro_EXI_CI}$ below.

ARs with a unit root of finite type

This section introduces the class of $\mathcal{H}$-valued ARs that is studied in the present paper, called ARs with a unit root of finite type. It also presents an existence result about their common trends representation, which shows that the solution of an AR with a unit root of finite type\ is necessarily $I(d)$ for some finite integer $d$ and displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks.\footnote{This result is a direct consequence of a well known theorem in operator theory, reported in Theorem $\ref{thm_FLSF}$ in Appendix $\ref{app_operator}$, and first employed in BS:18 in the context of $\mathcal{H}$-valued ARs with compact operators.} The relations of ARs with a unit root of finite type\ with the ARs studied in literature are also discussed in this section, and an example of an AR with a unit root of finite type\ with a non-compact AR operator is given.

Main assumption

Consider an $\mathcal{H}$-valued AR process

where $I$ indicates the identity operator in $\mathcal{L}_\mathcal{H}$, $\{\varepsilon_{t},t\in \mathbb{Z}\}$ is white noise and the operator function $$ A(z)=I-\sum_{h=1}^kA^\circ_h z^h,\qquad z \in \mathbb{C},\qquad A(1)\neq 0, $$ is non-invertible at $z=1$ and invertible in the punctured disc $D(0,\rho)\setminus\{1\}$ for some $\rho>1$.

This requirement restricts attention to unit roots at frequency zero, corresponding to the point $z=1$ on the unit disc. Note that there is no loss of generality in assuming that $A(1)\neq 0$. In fact, if $A(1)=0$, one can factorize $(1-z)^s$ from $A(z)$, $A(z)=(1-z)^s\widetilde{A}(z)$ for some $\widetilde{A}(1)\neq \{ 0 \}$ and some $s>0$, and rewrite the AR equations $A(L)x_{t}=\varepsilon_{t}$ as $\widetilde{A}(L)y_{t}=\varepsilon_{t}$ for $y_{t}=\Delta^s x_{t}$.

In order to state the key assumption, it is useful to expand the operator function $A(z)=I-\sum_{h=1}^k A^\circ_h z^h$ around 1, obtaining

where empty sums are defined to be 0 and hence $A_n = 0 $ for $n>k$.

The notion of eigenvalue of finite type, see GGK:90:1, is central in the present setup and it is reported next. For any $A \in\mathcal{L}_\mathcal{H}$ the subspace $\{v\in \mathcal{H}:Av=0 \}$, written $\operatorname{Ker} A$, is called the kernel of $A$ and the subspace $\{Av : v\in \mathcal{H}\}$, written $\operatorname{Im} A$, is called the image of $A$. The dimension of $\operatorname{Im} A$, written $\dim \operatorname{Im} A$, is called the rank of $A$.

Direct consequences of this definition are listed in the following remark.

The key assumption is introduced next.

That is, an AR with a unit root of finite type\ is such that $A(z)$ is invertible for all $z\in D(0,\rho)\setminus\{1\}$ for some $\rho>1$, $0<\dim \operatorname{Ker} A_0=\dim (\operatorname{Im} A_0)^\bot<\infty$ and $\operatorname{Im} A_0$ is closed, where $A_0$ is as in (ref).

Existence of a common trends representation

Under Assumption $\ref{ass_finite_type}$, one can apply the results in Section XI.9 of GGK:90:1, reported in Theorem $\ref{thm_FLSF}$ in Appendix $\ref{app_operator}$, and first employed in BS:18 in the context of $\mathcal{H}$-valued ARs with compact operators. These results guarantee that there exist a finite integer $d=1,2,\dots$ and finite rank operators $C_{0},C_{1},\dots,C_{d-1}$ such that

so that the inverse of $A(z)$ has a pole of finite order $d$ at $z=1$.

This implies that the solution of the AR equations is $I(d)$ for some finite integer $d$. Moreover, because the operators that make up the principal part of $A(z)^{-1}$ around $z=1$ have finite rank, $x_t$ displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks, as reported in Theorem $\ref{thm_EXI}$ below.

In order to state Theorem $\ref{thm_EXI}$, the cumulation operator $\mathcal{S}$ in introduced, following Gre:99.

Remark that by definition $\mathcal{S} $ assigns value 0 to the cumulated process at time 0. In fact, applying the definition, also see Properties 2.1, 2.2 in Gre:99, one has

Eq. (ref) shows that $\mathcal{S} $ applied to $\Delta w_{t}$ regenerates the level of the process $w_{t}$, up to a constant; this parallels the constant of integration in indefinite integrals. The integral operator $\mathcal{S} $ is hence the inverse of the difference operator $\Delta$ up a constant, which is set by Definition $\ref{def_S}$ so as to make the cumulated process $\mathcal{S} \Delta w_{t}$ equal to 0 at time 0.

Note that when $w_t=\varepsilon_t$ is white noise, (ref) implies that $s_{1,t}=\mathcal{S} \varepsilon_t$ is a bilateral $\mathcal{H}$-valued random walk, see Bos:00; because $\Delta s_{1,t}=\Delta \mathcal{S} \varepsilon_t=\varepsilon_{t}$ is $I(0)$, this shows that $s_{1,t}$ is $I(1)$. Similarly, for $h=2,3,\dots$, $s_{h,t}=\mathcal{S} s_{h-1,t}\sim I(h)$ is the $(h-1)$-fold cumulation of the bilateral random walk $s_{1,t}\sim I(1)$.

The following results connects ARs with a unit root of finite type with the existence of a common trend representation in terms of stochastic trends of the above type.

In the common trends representation (ref) the operators $C_{0},C_{1},\dots,C_{d-1}$ have finite rank; this implies that $x_t$ depends only on a finite number of bilateral (cumulated) random walks. In fact, these common stochastic trends are selected from $s_{h,t} \sim I(h)$, $h=1,\dots,d$, by the finite rank operators $C_{0},C_{1},\dots,C_{d-1}$ that load onto $x_t$ only a finite number of characteristics from $s_{h,t}$, $h=1,\dots,d$.

Theorem $\ref{thm_EXI}$ implies a number of properties for ARs with a unit root of finite type, some of which are listed in the following corollary.

Corollary (ref) lists some implications of Theorem $\ref{thm_EXI}$, namely that $d$ (the order of the pole of the inverse of $A(z)$ at $z=1$) is finite, the process is cointegrated, the number of common trends is finite and the number of cointegrating relations is infinite.

Despite these interesting implications of Theorem $\ref{thm_EXI}$, these existence results do not address a number of important issues, such as the connection between the structure of $A(z)$ and the order of integration $d$ of the process. In fact, one cannot determine the order of integration of the solution of the AR equations using Theorem $\ref{thm_EXI}$. Moreover, Theorem $\ref{thm_EXI}$ does not specify the connection between $\operatorname{Im} C_0$ and the AR operators, so that one does not know how to construct the attractor space\ and the cointegrating space\ in terms of the AR operators. Finally, the relations among the finite rank operators $C_{0},C_{1},\dots,C_{d-1}$ are not specified and hence Theorem $\ref{thm_EXI}$ is silent about the structure of the cointegrating relations.

These additional characterization results form the main contribution of the present paper; they go beyond Theorem $\ref{thm_EXI}$ and Corollary $\ref{coro_EXI_CI}$, and they are presented in full generality in Section $\ref{sec_char_d}$ for the generic $I(d)$ case. For ease of presentation, Section $\ref{sec_char_1_2}$ starts with the $I(1)$ and $I(2)$ cases.

Relations with the literature

Before turning to these results, the present section discusses the relationship between Assumption $\ref{ass_finite_type}$ and the assumptions employed in the literature. An example in the next section illustrates the differences.

The following proposition discusses the relation with CKP:16, who study $I(1)$ processes $x_t$ satisfying $\Delta x_t=B(L)\varepsilon_{t}$, where $\sum_{n=1}^{\infty}n\|B_{n}\|_{\mathcal{L}_\mathcal{H}}<\infty$ and $\dim \operatorname{Im} B(1)< \infty$.

This shows that $I(1)$ ARs with a unit root of finite type\ necessarily satisfy Assumption 2.1 in CKP:16; hence their asymptotic analysis applies and their test can be employed in the present setup.

The next proposition discusses the relation with HP:WP, who consider (ref) with $k=1$ and compact $A^\circ_1$. Similarly, BSS:17 consider (ref) with compact $A^\circ_1,\dots,A^\circ_k$ if $k>1$ and BS:18 consider (ref) with compact $A^\circ_1,\dots,A^\circ_k$ for $k\geq 1$.

This shows that the present results can be applied to the setups of HP:WP, BSS:17 and BS:18. BSS:17 also consider $x_t=A_1^{\circ}x_{t-1}+\varepsilon_t$ with no compactness assumption on $A_1^{\circ}$, see Proposition $\ref{prop_BSS_k_1}$ below.

Finally, CHP:WP consider an error correction form with compact error correction operator and show that in this case the number of $I(1)$ common trends is infinite and the dimension of the cointegrating space\ is finite. This case is not covered by the present results.

Example of a non-compact operator

This section illustrates the relevance of Assumption $\ref{ass_finite_type}$ with a simple example. This example is considered again in Section $\ref{sec_example}$ to illustrate the characterization results in the $I(1)$ case.

Consider $x_{t}=A^\circ_1 x_{t-1}+\varepsilon_{t}$ where $A^\circ_1$ is a band operator. Band operators are defined as follows: let $\varphi_1,\varphi_2,\dots$ be an orthonormal basis of $\mathcal{H}$ and let $(a_{ij})$, where $a_{ij}=\langle A\varphi_j,\varphi_i \rangle$, be the matrix representation of $A\in\mathcal{L}_\mathcal{H}$ corresponding to $\varphi_1,\varphi_2,\dots$, see e.g. GGK:03; $A\in\mathcal{L}_\mathcal{H}$ is called a band operator if all nonzero entries in its matrix representation $(a_{ij})$ are in a finite number of diagonals parallel to the main diagonal, i.e. there exists an integer $N$ such that $a_{ij}=0$ if $|i-j|>N$, see e.g. GGK:03.

Note that a band operator is compact if and only if $\lim_{i,j\rightarrow\infty}a_{ij}=0$, see Theorem 16.4 in GGK:03. Here $\lim_{i,j\rightarrow\infty}a_{ij}=0$ is not assumed, hence the operator $A^\circ_1$ is non-necessarily compact. Finally, let $z_{i,t}=\langle \varphi_i,z_t \rangle$ be the $i$-th coordinate of the process $z_t=x_t,\varepsilon_t$, and note that from (ref) one has $A_0 =I- A^\circ_1$, $A_1=A^\circ_{1}$ and $A_n=0$ for $n=2,3,\dots$, in $A(z)=\sum_{n=0}^{\infty}A_{n}(1-z)^n$, so that $A(z)=A_{0}+A_{1}(1-z)$.

Let $(a_{ij})$ be the matrix representation of $A^\circ_1$ and assume that $a_{ij}=0$ for $|i-j|>0$ and $a_{ii}=\alpha_i$, where $\alpha_i\in\mathbb{R}$, $\alpha_1=1$ and $0<|\alpha_i|<1$, $i=2,3,\dots$, so that $A^\circ_1$ is a band operator. Observe that $x_{t}=A^\circ_1 x_{t-1}+\varepsilon_{t}$ reads $$ x_{1,t} = x_{1,t-1} + \varepsilon_{1,t},\qquad \qquad x_{i,t} = \alpha_i x_{i,t-1} + \varepsilon_{i,t}, \qquad \qquad i=2,3,\dots. $$ Remark that $A^\circ_1$ is not compact because $\lim_{i,j\rightarrow\infty}a_{ij}=0$ is not imposed. Next note that $A(z)$ is invertible for all $z\in D(0,\rho)\setminus\{1\}$ for some $\rho>1$ and consider the matrix representation of $A^\circ_{1}=A_1$ and $A_0 =I- A^\circ_1$, i.e.

where empty entries are equal to 0, and compute $$ (\operatorname{Im} A_0)^\bot=(\overline{\operatorname{sp}}{\{\varphi_2,\varphi_3,\dots\}})^\bot=\operatorname{sp}{\{\varphi_1\}},\qquad \operatorname{Ker} A_0=\operatorname{sp}{\{\varphi_1\}}, $$ where $\operatorname{sp}\{\cdot\}$ and $\overline{\operatorname{sp}}\{\cdot\}$ indicate the span of the set of vectors in curly brackets and its closure respectively. Because $0<\dim \operatorname{Ker} A_0=\dim (\operatorname{Im} A_0)^\bot<\infty$, this shows that $A_0$ is Fredholm of index 0, so that Assumption $\ref{ass_finite_type}$ holds and $x_{t}=A^\circ_1 x_{t-1}+\varepsilon_{t}$ is an AR with a unit root of finite type\ with non-compact operator.

A characterization of $I(1)$ and $I(2)$ ARs with a unit root of finite type

This section presents a characterization of $I(1)$ and $I(2)$ ARs with a unit root of finite type. The $I(1)$ case parallels the results in HP:WP,BSS:17,BS:18, and it is discussed in Theorem $\ref{thm_CHAR_1}$. The results for the $I(2)$ case are novel, and they are given in Theorem $\ref{thm_CHAR_2}$.

I(1) case

The following notation is employed: write $A(z)=\sum_{n=0}^{\infty}A_{n}(1-z)^n$ as in (ref) and define

where $P_\eta\in \mathcal{L}_\mathcal{H}$ indicates the orthogonal projection on $\eta$, i.e. $P_\eta^2=P_\eta$, $\operatorname{Im} P_\eta=\eta$ and $\operatorname{Ker} P_\eta=\eta^\bot$.

Observe that $$ \zeta_1\subseteq \zeta_0^\bot,\qquad \tau_1\subseteq \tau_0^\bot $$ by construction; that is, $\zeta_1$ is orthogonal to $\zeta_0$ and $\tau_1$ is orthogonal to $\tau_0$. Moreover, because 1 is an eigenvalue of finite type, one has $0<\dim \tau_0^\bot=\dim \zeta_0^\bot<\infty$, see Remark $\ref{rem_eigFT0}$, so that the subspaces $\zeta_1,\tau_1$ are finite dimensional. In the following, $a \Rightarrow b$ indicates that $a$ implies $b$ and the orthogonal direct sum decomposition

is called the pole{$(1)$} condition.

Theorem $\ref{thm_CHAR_1}$ shows that an AR with a unit root of finite type\ generates an $I(1)$ process if and only if $\tau_1 = \tau_0^\bot\neq \{ 0 \}$. The common trends representation of $x_t$ shows that the $I(1)$ stochastic trends $s_{1,t}$ are loaded into the process by $C_0$; because $\operatorname{Im} C_0$ coincides with $\tau_1$, $\tau_1$ is the finite dimensional attractor space\ and the number of $I(1)$ trends in $x_t$ is finite and equal to $\dim \tau_1$.

Moreover, because $\operatorname{Im} C_0=\tau_1 = \tau_0^\bot$, for any nonzero $v \in \tau_0$ one has $\langle v,C_0 y \rangle = 0$ for all $y\in \mathcal{H}$, and hence also for $y=s_{d,t}$ in (ref); this implies that $\langle v,x_t\rangle$ is stationary, i.e. $\tau_0$ is the infinite dimensional cointegrating space. Note that this decomposition is orthogonal.

Using orthogonal projections, one can see that this orthogonal direct sum decomposition can be employed in general to characterize the degree of integration of any $v$-characteristic of the process. In fact, note that (ref) implies $P_{\tau_0}+P_{\tau_1}=I$, where $P_{\tau_h}$ is the orthogonal projection onto $\tau_h$; hence for any nonzero $v\in \mathcal{H}$ one has $\langle v,x_t\rangle=\langle v_0,x_t\rangle+\langle v_1,x_t\rangle$, where $v_h=P_{\tau_h}v\in \tau_h$, so that (ref) and (ref) describe the order of integration of any nonzero $v$-characteristic $\langle v,x_t\rangle$ of $x_t$. In particular one has $\langle v,x_t\rangle\sim I(1)$ if and only if $v_1\neq 0$, because $\langle v_0,x_t\rangle$ is $I(0)$.

Theorem (ref) further shows that for any nonzero $ v\in \tau_0$, $\langle v,x_t\rangle$ is not only stationary, but $I(0)$. This echoes the finite dimensional case, see Theorem 4.2 in Joh:96, except for the fact that the number of $I(0)$ cointegrating relations is infinite.

The pole{$(1)$} condition is next compared to equivalent conditions in the literature. BSS:17 define the following non-orthogonal direct sum decomposition

where $A_0,A_1$ are as in (ref); they call (ref) the `Johansen $I(1)$ condition'. Their Theorem 4.1 assumes that $x_t=A_1^{\circ}x_{t-1}+\varepsilon_t$ with no compactness assumption on $A_1^{\circ}$ and that the $k=1$ version of (ref) holds, i.e. $\mathcal{H}=\operatorname{Im} A_0 \oplus \operatorname{Ker} A_0$. Under these conditions, they finds the common trends representation (ref) with $d=1$ and $\operatorname{Im} C_0 = \operatorname{Ker} A_0$. The following proposition clarifies the connection between ARs with a unit root of finite type\ and their result.

One can observe that the case with infinite dimensional $\operatorname{Ker} A_0$, which correspons to an infinite dimensional attractor space, is not covered by the present results.

Finally, the following proposition proves the equivalence of the orthogonal direct sum condition in (ref) and the nonorthogonal direct sum conditions in (ref).

I(2) case

The $I(2)$ case is considered next. Consider $A(z)=\sum_{n=0}^{\infty}A_{n}(1-z)^n$ in (ref), and let $\zeta_0,\tau_0$ be as in (ref), consider $\zeta_1,\tau_1$ as in (ref), and define

where $\mathscr{Z}_{2}=\zeta_0\oplus\zeta_1$, $\mathscr{T}_{2}=\tau_0\oplus\tau_1$ and $A_{2,1}=A_2-A_1A_0^+A_1$, where the generalized inverse $A_0^+$ exists and it is unique, see Remark $\ref{rem_eigFT0}$.

Observe that $$ \zeta_2\subseteq (\zeta_0\oplus\zeta_1)^\bot,\qquad \tau_2\subseteq (\tau_0\oplus\tau_1)^\bot $$ by construction; that is, for $0< j<h$, $\zeta_h$ is orthogonal to $\zeta_j$, and $\tau_h$ is orthogonal to $\tau_j$. Moreover, because $0<\dim \zeta_0^\bot = \dim \tau_0^\bot < \infty$, the subspaces $\zeta_2,\tau_2$ are finite dimensional. In the following, the orthogonal direct sum decomposition

is called the pole{$(2)$} condition.

Some remarks on Theorem $\ref{thm_CHAR_2}$ are in order.

Illustrations

This section illustrates Theorems $\ref{thm_CHAR_1}$ and $\ref{thm_CHAR_2}$ via two simple examples, called the $I(1)$ and the $I(2)$ examples.

$I(1)$ example. Consider the setup in Section $\ref{sec_example_intro}$. Here the analysis should deliver that $x_t$ is $I(1)$, the attractor space\ coincides with $\operatorname{sp}\{\varphi_1\}$ and the cointegrating space\ with $\overline{\operatorname{sp}}\{\varphi_2,\varphi_3,\dots\}$. Since $\langle v,x_t\rangle$ is $I(0)$ for any nonzero $v\in \overline{\operatorname{sp}}\{\varphi_2,\varphi_3,\dots\}$ and $\langle v,x_t\rangle$ is $I(1)$ for any nonzero $v\in \operatorname{sp}\{\varphi_1\}$, the analysis should further convey that $\tau_0=\overline{\operatorname{sp}}\{\varphi_2,\varphi_3,\dots\}$ and $\tau_1=\operatorname{sp}\{\varphi_1\}$.

From (ref), one has

This shows that $\mathcal{H}=\tau_0 \oplus \tau_1$, $\tau_1\neq \{ 0 \}$, so that the pole{$(1)$} condition in (ref) holds and Theorem $\ref{thm_CHAR_1}$ applies: the common trends representation of $x_t$ is found by setting $d=1$ in (ref), $\operatorname{Im} C_0=\tau_1=\operatorname{sp}{\{\varphi_1\}}$ is the finite dimensional attractor space\ and $\tau_0=\overline{\operatorname{sp}}{\{\varphi_2,\varphi_3,\dots\}}$ is the infinite dimensional cointegrating space.

$I(2)$ example. Let $(a_{ij})$ be the matrix representation of $A^\circ_1$ and assume that $a_{ij}=0$ for $|i-j|>1$, $a_{12}=1$ and $a_{ii}=\alpha_i$, where $\alpha_i\in\mathbb{R}$, $\alpha_1=\alpha_2=\alpha_3=1$ and $0<|\alpha_i|<1$, $i=4,5,\dots$. Again here, $A^\circ_1$ is not necessarily compact but $x_{t}=A^\circ_1 x_{t-1}+\varepsilon_{t}$ is an AR with a unit root of finite type, as shown below. Observe that $x_{t}=A^\circ_1 x_{t-1}+\varepsilon_{t}$ reads $$

$$ Hence the analysis should deliver that $x_t$ is $I(2)$, the attractor space\ coincides with $\operatorname{sp}\{\varphi_1\}$ and the cointegrating space\ with $\overline{\operatorname{sp}}\{\varphi_2,\varphi_3,\dots\}$. Next note that $\langle v,x_t\rangle$ is $I(0)$ for any nonzero $v\in \overline{\operatorname{sp}}\{\varphi_4,\varphi_5,\dots\}$ and $\langle v,x_t\rangle$ is $I(1)$ for any nonzero $v\in \operatorname{sp}\{\varphi_2,\varphi_3\}$. Moreover, because $\Delta x_{1,t} = x_{2,t-1} + \varepsilon_{1,t} = x_{2,t} - \varepsilon_{2,t} + \varepsilon_{1,t}$, one has that $x_{2,t} - \Delta x_{1,t}$ is $I(0)$, i.e. $\langle \varphi_2 , x_{t}\rangle - \langle \varphi_1 , \Delta x_{t}\rangle$ is $I(0)$, so that $\langle \varphi_2 , x_{t}\rangle$ allows for polynomial cointegration while $\langle \varphi_3 , x_{t}\rangle$ does not. Hence the analysis should further convey that $\tau_0=\overline{\operatorname{sp}}\{\varphi_2,\varphi_4,\varphi_5,\dots\}$, $\tau_1=\operatorname{sp}\{\varphi_3\}$, $\tau_2=\operatorname{sp}\{\varphi_1\}$, $\langle \varphi_2,A_0^+A_1\Delta x_{i,t}\rangle=-\Delta x_{i,t}$, and $\langle \varphi_i,A_0^+A_1\Delta x_{i,t}\rangle=0$ for $i=4,5,\dots$.

Consider the matrix representation of $A^\circ_{1}=A_1$ and $A_0 =I- A^\circ_1$, i.e. $$ A^\circ_1=A_1=\left(

\right),\qquad A_0=\left(

\right), $$ where empty entries are equal to 0. Compute $$ \zeta_0=\operatorname{Im} A_0=\overline{\operatorname{sp}}{\{\varphi_1,\varphi_4,\varphi_5,\dots\}},\qquad \tau_0=(\operatorname{Ker} A_0)^\bot=(\operatorname{sp}{\{\varphi_1,\varphi_3\}})^\bot=\overline{\operatorname{sp}}{\{\varphi_2,\varphi_4,\varphi_5,\dots\}}, $$ so that $\zeta_0^\bot=\operatorname{sp}{\{\varphi_2,\varphi_3\}}$ and $\tau_0^\bot=\operatorname{sp}{\{\varphi_1,\varphi_3\}}$; because $0<\dim \operatorname{Ker} A_0=\dim (\operatorname{Im} A_0)^\bot<\infty$, this shows that $A_0$ is Fredholm of index 0\ and because $A(z)=I-A^\circ_1 z$ is invertible for all $z\in D(0,\rho)\setminus\{1\}$ for some $\rho>1$, $x_{t}=A^\circ_1 x_{t-1}+\varepsilon_{t}$ is an AR with a unit root of finite type\ with non-compact operator.

Next compute $$ \zeta_1=\operatorname{Im} P_{\zeta_0^\bot}A_{1}P_{\tau_0^\bot} = \operatorname{sp}{\{\varphi_3\}},\qquad \tau_1=(\operatorname{Ker} P_{\zeta_0^\bot}A_{1}P_{\tau_0^\bot})^\bot=(\overline{\operatorname{sp}}{\{\varphi_1,\varphi_2,\varphi_4,\varphi_5,\dots\}})^\bot=\operatorname{sp}{\{\varphi_3\}}. $$ This shows that $\tau_1\subset \tau_0^\bot$, so that the pole{$(1)$} condition in (ref) does not hold and the process is $I(d)$ for some finite $d=2,3,\dots$.

Now consider $P_{\mathscr{Z}_{2}^\bot}A_{2,1}P_{\mathscr{T}_{2}^\bot}$ in (ref); since $\mathscr{Z}_{2}=\zeta_0\oplus\zeta_1=\overline{\operatorname{sp}}{\{\varphi_1,\varphi_3,\varphi_4,\dots\}}$ and $\mathscr{T}_{2}=\tau_0\oplus\tau_1=\overline{\operatorname{sp}}{\{\varphi_2,\varphi_3,\dots\}}$, one has $\mathscr{Z}_{2}^\bot=\operatorname{sp}{\{\varphi_2\}}$ and $\mathscr{T}_{2}^\bot=\operatorname{sp}{\{\varphi_1\}}$. Note that $A_2=0$ and hence $A_{2,1}=-A_1A_0^+A_1$; thus $P_{\mathscr{Z}_{2}^\bot}A_{2,1}P_{\mathscr{T}_{2}^\bot}=-P_{\operatorname{sp}\{\varphi_2\}}A_1A_0^+A_1 P_{\operatorname{sp}\{\varphi_1\}}$ and because $P_{\operatorname{sp}\{\varphi_2\}}A_1=P_{\operatorname{sp}\{\varphi_2\}}$ and $A_1P_{\operatorname{sp}\{\varphi_1\}}=P_{\operatorname{sp}\{\varphi_1\}}$, one has $P_{\mathscr{Z}_{2}^\bot}A_{2,1}P_{\mathscr{T}_{2}^\bot}=-P_{\operatorname{sp}\{\varphi_2\}}A_0^+P_{\operatorname{sp} \{\varphi_1\}}$. Next the matrix representation of $A_0^+$ is investigated; from Lemma $\ref{lem_gen_inv_h}$ one has $\operatorname{Ker} A_0^+ =(\operatorname{Im} A_0)^\bot$, $A_0^+ A_0=P_{(\operatorname{Ker} A_0)^\bot}$ and because $(\operatorname{Im} A_0)^\bot=\operatorname{sp}{\{\varphi_2,\varphi_3\}}$ and $(\operatorname{Ker} A_0)^\bot=\overline{\operatorname{sp}}{\{\varphi_2,\varphi_4,\varphi_5,\dots\}}$ one has $$ A_0^+=\left(

\right). $$ This implies that the only nonzero element in the matrix representation of $P_{\mathscr{Z}_{2}^\bot}A_{2,1}P_{\mathscr{T}_{2}^\bot}=-P_{\operatorname{sp}\{\varphi_2\}}A_0^+P_{\operatorname{sp}\{\varphi_1\}}$ is a one in row 2 and column 1, so that $$ \zeta_2=\operatorname{Im} P_{\mathscr{Z}_{2}^\bot}A_{2,1}P_{\mathscr{T}_{2}^\bot}=\operatorname{sp}{\{\varphi_2\}},\qquad \tau_2=(\operatorname{Ker} P_{\mathscr{Z}_{2}^\bot}A_{2,1}P_{\mathscr{T}_{2}^\bot})^\bot=(\overline{\operatorname{sp}}{\{\varphi_2,\varphi_3,\dots\}})^\bot= \operatorname{sp}{\{\varphi_1\}}. $$ Hence $\mathcal{H}=\tau_0 \oplus \tau_1 \oplus \tau_2$, $\tau_2\neq \{ 0 \}$, i.e. the \textsc{pole}{$(2)$} condition in \eqref{eq_I2_cond} holds and Theorem $(ref)$ applies: the common trends representation of $x_t$ is found by setting $d=2$ in \eqref{eq_CT}, $\operatorname{Im} C_0=\tau_2=\operatorname{sp}{\{\varphi_1\}}$ is the finite dimensional attractor space\ and $\tau_0 \oplus \tau_1=\overline{\operatorname{sp}}{\{\varphi_2,\varphi_3,\dots\}}$ is the infinite dimensional cointegrating space. Moreover, for any nonzero $v\in\mathcal{H}$ one has

Note that $\tau_1\subset\tau_0^\bot$ and $\tau_2=(\tau_0\oplus\tau_1)^\bot\neq \{ 0 \}$. Moreover, $\langle \varphi_2,A_0^+A_{1}\Delta x_t\rangle=-\Delta x_{1,t}-\Delta x_{2,t}$ and $\langle \varphi_i,A_0^+A_{1}\Delta x_{t}\rangle=\frac{\alpha_i}{1-\alpha_i} \Delta x_{i,t}$, for $i=4,5,\dots$; hence $\langle \varphi_2,x_t\rangle + \langle \varphi_2,A_0^+A_{1}\Delta x_t\rangle= x_{2,t}-\Delta x_{1,t}-\Delta x_{2,t}$ and, for $i=4,5,\dots$, $\langle \varphi_i,x_t\rangle + \langle \varphi_i,A_0^+A_{1}\Delta x_t\rangle= x_{i,t} + \frac{\alpha_i}{1-\alpha_i} \Delta x_{i,t}$. This shows that $\langle v,x_t\rangle + \langle v,A_0^+A_{1}\Delta x_t\rangle$ contains stationary terms ($\Delta x_{2,t}$ and $\frac{\alpha_i}{1-\alpha_i} \Delta x_{i,t}$) that are not necessary for cointegration; these can be eliminated by considering $\langle v,A_0^+A_{1}P_{\tau_2}\Delta x_t\rangle$ instead of $\langle v,A_0^+A_{1}\Delta x_t\rangle$, as in the finite dimensional $I(2)$ case, see Theorem 4.6 in Joh:96.

A characterization of $I(d)$ ARs with a unit root of finite type

This section extends the results in Section $\ref{sec_char_1_2}$ to the general $I(d)$, $d=1,2,\dots<\infty$, case. Theorem $\ref{thm_CHAR_d}$ provides a necessary and sufficient condition for ARs with a unit root of finite type\ to be $I(d)$ and it is shown that under this condition the space $\mathcal{H}$ is decomposed into the direct sum of $d+1$ orthogonal subspaces $\tau_h$, $\mathcal{H}=\tau_0\oplus\tau_1\oplus\dots\oplus\tau_d$, $\tau_d\neq \{ 0 \}$, that are defined in terms of $A_0,A_1,\dots, A_d$ in (ref), see Definition $\ref{def_LRF}$ below.

The finite dimensional attractor space\ coincides with $\tau_d$ and $\tau_0\oplus \tau_1\oplus\dots\oplus\tau_{d-1}$ is the infinite dimensional cointegrating space. In $\tau_0$, which is infinite dimensional, one finds the cointegrating vectors that allow for polynomial cointegration of order 0 and in $\tau_h$, $h=1,\dots,d-2$, which is finite dimensional and can as well be equal to 0, those that allow for polynomial cointegration of order $h$. In $\tau_{d-1}$, with $ 0 \leq \dim \tau_{d-1} < \infty$, those that are $I(d-1)$ and don't allow for polynomial cointegration. Finally, any nonzero $v\in\tau_{d}$ is such that $\langle v,x_t \rangle \sim I(d)$. The results in Section $\ref{sec_char_1_2}$ are found as special cases for $d=1$ and $d=2$. Before stating the results, some definitions are introduced.

A few remarks are in order.

In the following, the orthogonal direct sum decomposition

is called the pole{$(d)$} condition.

As discussed in the next remark, the only difference with the finite dimensional case, see FP_idcoeff, is that in that case the number of cointegrating relations of order 0 is finite.

In order to complete the discussion of the relation of the present results with the existing literature, the equivalence of the pole{$(d)$} condition in (ref) to the condition in HP:WP reported in eq. (ref) below is discussed.

HP:WP consider $x_t=A_1^{\circ}x_{t-1}+\varepsilon_t$ with $A_1^{\circ}$ compact and formulate an $I(d)$ condition and then study the $I(1)$ case. In order to state their $I(d)$ condition, they employ the nonorthogonal direct sum decomposition $\mathcal{H}=\mathcal{H}_P \oplus \mathcal{H}_T$, where $\mathcal{H}_P$ is the finite dimensional image of the Riesz projection associated with the isolated eigenvalue $z=1$ and $\mathcal{H}_T$ is the infinite dimensional image of the Riesz projections associated with the remaining stable eigenvalues. Using the nonorthogonal projections associated to the nonorthogonal direct sum decomposition $\mathcal{H}=\mathcal{H}_P \oplus \mathcal{H}_T$, they decompose the process into $x_t=x_t^P+x_t^T$, where $x_t^X= A_X x_t^X+ \varepsilon_t^X \in \mathcal{H}_X$ and $A_X$ is the restriction of $A_1^{\circ}$ to $\mathcal{H}_X$, $X=T,P$. Their $I(d)$ condition is stated as

i.e. $(A_P-I)^{d-1}\neq0$ and $(A_P-I)^{d}=0$, which simplifies to $A_P=I$ in the $I(1)$ case studied in that paper.

Conclusion

The present paper characterizes the cointegration properties of ARs with a unit root of finite type, i.e. $\mathcal{H}$-valued AR processes $A(L)x_{t}=\varepsilon_{t}$ such that $A(z)$ has an eigenvalue of finite type at $z=1$ and it is invertible in the punctured disc $D(0,\rho)\setminus\{1\}$ for some $\rho>1$. It is shown that ARs with a unit root of finite type\ are necessarily integrated of finite order $d$ and necessarily have a finite number of $I(d)$ trends and an infinite dimensional cointegrating space. This is in line with the setup employed in most contributions in the literature and seems to be the most empirically relevant framework.

A necessary and sufficient condition on the AR operators that establishes the value of $d$ is given in terms of the orthogonal direct sum decomposition $\mathcal{H}=\tau_0\oplus\tau_1\oplus\dots\oplus\tau_d$, $\tau_d\neq \{ 0 \}$, where $\tau_0$ is infinite dimensional, $0 \leq \dim \tau_h < \infty$, $h=1,\dots,d-1$, with strict inequality for $h = d$.

A full description of how the properties of the characteristic $\langle v,x_t\rangle$ vary with $v\in \mathcal{H}$ is given: in $\tau_0$, one can combine $\langle v,x_t\rangle$ with differences of the process and find at most $I(0)$ polynomial cointegrating relations, in $ \tau_1$, one can combine $\langle v,x_t\rangle$ with differences and find at most $I(1)$ polynomial cointegrating relations, and so on up to $\tau_{d-2}$, in which one can combine $\langle v,x_t\rangle$ with differences and find at most $I(d-2)$ polynomial cointegrating relations. Finally, any nonzero $v\in\tau_{d-1}$ is such that $\langle v,x_t\rangle$ is $I(d-1)$ and does not allow for polynomial cointegration and any nonzero $ v\in \tau_{d}$ is such that $\langle v,x_t\rangle$ is $I(d)$. This shows that the infinite dimensional subspace $\tau_0\oplus\tau_1\oplus\dots\oplus\tau_{d-1}$ is the cointegrating space\ while the finite dimensional subspace $\tau_d$ is the attractor space.

For any nonzero $v$ in the cointegrating space, the expression of the polynomial cointegrating relations is provided in terms of operators that are defined recursively in terms of the AR operators together with the $\tau_h$.

The present results show that, under the assumption that 1 is an eigenvalue of finite type of the AR operator function, the infinite dimensionality of the space does not introduce additional elements in the analysis. That is, apart from the fact that the number of $I(0)$ cointegrating relations is infinite, conditions and properties of $\mathcal{H}$-valued AR processes coincide with those that apply in the usual finite dimensional VAR case.