Representation of I(1) and I(2) autoregressive Hilbertian processes

Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.

67,908 characters · 5 sections · 80 citation commands

Rendered from LaTeX for readability, not typeset faithfully. Citation keys are highlighted; maths is left as source; figures, tables and equation environments are summarised rather than reproduced; unrecognised commands are greyed out so nothing is silently dropped. Email addresses are removed.

\affil[1]{School of Economics, University of Sydney} \affil[2]{Department of Economics, Queen's University}

center[center omitted — 71 chars of source]

abstractWe develop versions of the Granger-Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.

\onehalfspacing

Introduction

Results on the existence and representation of integrated solutions to vector autoregressive laws of motion are among the most important and subtle contributions of econometricians to time series analysis, yet also among the most widely misunderstood. The best known such result is the so-called Granger representation theorem, which first appeared in an unpublished UC San Diego working paper of G83. In this paper, Granger, having recently introduced the concept of cointegration G81 sought to connect statistical models of time series based on linear process representations to regression-based models involving equilibrium correction terms, which had appeared in work by S64 and DHSY78. The main result of G83 first emerged in published form in G86 without proof, but more prominently in the widely cited Econometrica article by EG87, where it is labeled the “Granger representation theorem”, with the exclusion of the first author presumably due to the paper having resulted from the merger of previous independent contributions.

The proof of the Granger representation theorem in EG87 is incorrect. Moreover, the error can be traced back to the original working paper of G83. A counterexample to Lemma A1 of EG87, which is also Theorem 1 of G83, may be found buried in a footnote of J09. Johansen was familiar with Granger's work on representation theory at an early stage, visiting UC San Diego and authoring a closely related Johns Hopkins working paper in 1985 that was eventually published as J88. At around the same time the doctoral thesis of Y87 at UC San Diego established the connection to Smith-McMillan forms. J91 provided what appears to be the first correct statement and proof of a modified version of the Granger representation theorem, which we will call the Granger-Johansen representation theorem. This contribution did not merely correct a technical error of Granger; it reoriented attention toward a central issue: when does a given vector autoregressive law of motion admit an I(1) solution? The answer to this question is given by the Johansen I(1) condition, which is a necessary and sufficient condition on the autoregressive polynomial and its first derivative at one for a vector autoregressive law of motion to admit an I(1) solution. J92 subsequently obtained analogous results for I(2) processes, and L98 provided some results for higher-order integrated processes.

An early contribution of S91 contained a striking observation on the Johansen I(1) condition: it corresponds to a necessary and sufficient condition for the inverse of a matrix pencil to have a simple pole at a given point in the complex plane. Various authors later exploited this insight, including FZ02,FZ09,FZ11, F07, J09 and FP11,FP16. A nice aspect of the connection to analytic function theory is that it extends naturally to the development of $\mathrm{I}(d)$ representation theory with integer-valued $d\geq2$: just as the Johansen I(1) condition can be reformulated as a necessary and sufficient condition for a simple pole, analogous $\mathrm{I}(d)$ conditions can be reformulated as necessary and sufficient conditions for poles of order $d$. FP19 have recently taken precisely this approach to develop a general $\mathrm{I}(d)$ representation theory. The introduction to their paper contains a detailed discussion of the history of research on the Granger-Johansen representation theorem.

Parallel to the development of representation theorems for cointegrated systems in the 1980s and early 1990s was the development of asymptotic distribution theory for the statistical estimation of such systems, obtained by applying central limit theory on function spaces and associated results. This research was led by Phillips and his students at Yale; see, in particular, PD86, P86,P88,P91, PPk88, PkP88,PkP89 and PH90, among many other contributions. J91 used limit theorems developed in this body of work, applicable to integrated systems exhibiting general serial dependence, to derive the asymptotic distribution of the maximum likelihood estimator for I(1) Gaussian vector autoregressive systems. Complementary work by CW87,CW88 and KP91 on asymptotic theory for regression with integrated processes has also been influential.

In this paper we provide representation theorems for I(1) and I(2) vector autoregressive processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series HK12, and was first studied by CKP16 in the case of I(1) probability density-valued time series; see also B17 and SB19. Our results here build on those we obtained in an earlier paper with J.\ Seo BSS17 establishing a representation theorem for the I(1) case. They differ from our earlier results in their explicit use of analytic function theory. In Theorems (ref) and (ref), our I(1) and I(2) representation theorems, we directly assume that the inverse of the autoregressive operator pencil has a pole of first or second order at one. We elaborate upon the meaning of these pole conditions in Theorems (ref) and (ref), which provide necessary and sufficient conditions to have a pole of first or second order. These results also provide explicit formulas for the coefficients in the principal part of the corresponding Laurent series.

Our paper supersedes an earlier manuscript posted on the arXiv.org preprint repository in January 2017 BS17 that dealt only with the I(1) case. During its preparation several working papers have emerged that deliver related results. In particular, FP18 study I($d$) solutions to autoregressive laws of motion in complex separable Hilbert space, for integer-valued $d\geq1$. Their necessary and sufficient condition for an I($d)$ solution involves an orthogonal direct sum decomposition of the Hilbert space into $d$ closed subspaces. This contrasts with the direct sum conditions given by BSS17 for the I(1) case, and here for the I(1) and I(2) cases, which involve nonorthogonal direct sums. We also provide a range of alternative formulations of our necessary and sufficient conditions, some of which may be easier to verify than others. Also relevant is recent work by HP17, who established an alternative I($d$) condition for first-order autoregressive Hilbertian processes: the restriction of the autoregressive operator to the image of the Riesz projection associated with its unit eigenvalue differs from the identity by an operator nilpotent of degree $d$. Finally, CHP16 have developed I(1) representation theory for autoregressive Hilbertian processes under the assumption that the impact operator in the error correction representation is compact. Under this condition the dimension of the cointegrating space must be finite, which contrasts with the setting of this paper and the others cited in this paragraph, where the codimension of the cointegrating space must be finite. Finite codimensionality of the cointegrating space implies that the I(1) stochastic trend in the Beveridge-Nelson representation of our cointegrated process is confined to a finite dimensional linear subspace. It is a consequence of a compactness condition we impose on the autoregressive operators. FP18 have observed that finite codimensionality of the cointegrating space holds more generally if the autoregressive operator pencil has an eigenvalue of finite type at one.

We structure the remainder of the paper as follows. Section (ref) sets the scene with notation and essential mathematics. Our results on I(1) and I(2) processes are contained in Sections (ref) and (ref) respectively. We provide a brief discussion of directions for future research in Section (ref). Appendix (ref) contains background material on the spectral properties of operator-valued functions, including a statement of the analytic Fredholm theorem, which is a key input to our results. The proofs of our results are collected in Appendix (ref).

Preliminaries

The setting for our analysis is a separable complex Hilbert space $\mathcal H$ with inner product $\langle\cdot{,}\cdot\rangle$ and norm $\Vert\cdot\Vert$. If $\mathcal H' $ is another such space, we let $\mathcal L_{\mathcal H,\mathcal H'}$ denote the Banach space of continuous linear operators from $\mathcal H$ to $\mathcal H'$ equipped with the operator norm. We are mostly concerned with the case $\mathcal H=\mathcal H'$, and write $\mathcal L_{\mathcal H}$ in place of $\mathcal L_{\mathcal H,\mathcal H}$. To each operator $A\in\mathcal{L}_{\mathcal H}$ we associate two linear subspaces of $\mathcal H$: the kernel and range of $A$, given by

equation*[equation* omitted — 104 chars of source]

We let $\mathrm{I}\in\mathcal{L}_{\mathcal H}$ denote the identity map on $\mathcal H$.

A central concern of our analysis will be the decomposability of $\mathcal H$ into sums of certain linear subspaces of $\mathcal H$. Given linear subspaces $V$ and $W$ of $\mathcal H$, we write $V+W$ for the linear subspace of all $x\in\mathcal H$ such that $x=v+w$ for some $v\in V$ and $w\in W$. When $V$ and $W$ are linear subspaces of $\mathcal{H}$ with $V\cap W=\{0\}$, we may instead write $V\oplus W$ for their sum, and call it a direct sum. When we write $\mathcal H=V\oplus W$, we are asserting that $\mathcal H=V+W$ and that $V\cap W=\{0\}$. In this case, any $x\in\mathcal H$ may be uniquely decomposed as $x=v+w$ with $v\in V$ and $w\in W$.

Orthogonal complements and projections play a key role in our analysis. Given a linear subspace $V$ of $\mathcal H$, we define its orthogonal complement by

equation*[equation* omitted — 89 chars of source]

The orthogonal complement to a linear subspace of $\mathcal H$ is always a closed linear subspace of $\mathcal H$. Given a closed linear subspace $V$ of $\mathcal H$, it is always the case that $\mathcal H=V\oplus V^\perp$. Thus any $x\in\mathcal H$ may be uniquely decomposed as $x=v+v'$ with $v\in V$ and $v'\in V^\perp$. We denote by $\mathrm{P}_V\in\mathcal{L}_{\mathcal H}$ the orthogonal projection on $V$, which maps a point $x=v+v'$ to $v$.

Our main results concern the representation of time series taking values in $\mathcal H$, but only the most basic understanding of probability on $\mathcal H$ is required. As in BSS17, we let $L_{\mathcal H}^2$ denote the Banach space of random elements $Z$ of $\mathcal H$ (identifying random elements that are equal with probability one) that satisfy $E\Vert Z\Vert^2<\infty$ and $EZ=0$, equipped with the norm $\Vert Z\Vert_{L^2_{\mathcal H}}=(E\Vert Z\Vert^2)^{1/2}$. Refer to that paper for the definition of $EZ$ and of the covariance operator of an element of $L_{\mathcal H}^2$. For further details, the monograph of B00 provides a comprehensive treatment of linear processes taking values in a real Hilbert or Banach space. A complex Hilbert space setting was studied more recently by CH17.

I(1) autoregressive Hilbertian processes

In this section we state our results for I(1) autoregressive processes. Let $p \in \mathbb{N}$, and consider the following AR($p$) law of motion in $\mathcal{H} $:

equation[equation omitted — 75 chars of source]

Here, the $X_t$'s and $\varepsilon_t$'s are random elements of $\mathcal{H}$, and the $\Phi_j$'s are continuous linear operators from $\mathcal{H}$ to $\mathcal{H}$. We say that the AR($p$) law of motion (ref) is engendered by the map $\Phi : \mathbb{C} \mapsto \mathcal L_\mathcal{H} $ given by

align[align omitted — 78 chars of source]

We will refer to an operator-valued polynomial function of a complex variable as an operator pencil; note that some authors reserve this term for linear polynomials. We impose the following conditions on the objects just introduced.

assumption(i) $\varepsilon = (\varepsilon_t, t \in \mathbb{Z})$ is an iid sequence in $ L^2_\mathcal{H}$ with positive definite covariance operator $ \Sigma \in \mathcal L_\mathcal{H}$. (ii) $ \Phi_1, \ldots, \Phi_p$ are compact operators in $\mathcal L_\mathcal{H}$ such that $ \Phi : \mathbb{C} \mapsto \mathcal L_\mathcal{H}$ is noninvertible at $z=1 $ and invertible at every other $z$ in the closed unit disk.

remarkThe innovations $\varepsilon_t$ are referred to as strong white noise due to their being centered (i.e.\ zero expected value) and iid. We impose these conditions for simplicity, but the results to be developed remain valid if the iid condition is replaced with the weaker requirement that the cross-covariance operators for the $\varepsilon_t$'s are all zero, as in FP18. In the latter case the $\varepsilon_t$'s are merely said to be white noise. More generally, one might consider allowing the $\varepsilon_t$'s to be a general I(0) process as in J92 and CP09,CP12, or even a subexponential process as in A18, but we do not pursue this route here.

remarkThe results to be developed remain valid if the assumption that $\Phi_1,\ldots,\Phi_p$ are compact is replaced with the weaker but less easily interpretable requirement that $\Phi(z)$ has an eigenvalue of finite type at $z=1$. See FP18 for details.

The approach we will take to developing representation theory for I(1) and I(2) autoregressive processes in $\mathcal H$ essentially boils down to studying the behavior of $\Phi(z)^{-1}$ near $z=1$. We achieve this by applying the analytic Fredholm theorem, a complete statement of which is provided in Appendix (ref). To apply this result we need $\Phi(z)$ to be analytic in $z$ (in fact, it is polynomial in $z$, hence analytic), and Fredholm operator-valued (meaning that it has finite dimensional kernel and cokernel for all $z$, which is guaranteed by our compactness condition on the $\Phi_j$'s). The analytic Fredholm theorem implies that, for all $z$ in a punctured neighborhood of one, we have

equation[equation omitted — 90 chars of source]

where $d\in\mathbb N$ and $\Upsilon_{-d},\Upsilon_{-d+1},\ldots$ is a sequence in $\mathcal L_\mathcal{H}$. The series in (ref) is called the Laurent series of $\Phi(z)^{-1}$ around $z=1$, and converges in $\mathcal L_{\mathcal H}$. If we assume without loss of generality that $\Upsilon_{-d}\neq0$, then $\Phi(z)^{-1}$ is said to have a pole of order $d$ at $z=1$. The operator $\Upsilon_{-1}$ is called the residue of $\Phi(z)^{-1}$ at $z=1$. A pole of order one is said to be simple. We call the sum of the leading terms indexed by $k=-d,\ldots,-1$ the principal part of the Laurent series, and we call the truncated series excluding these leading terms the analytic part of the Laurent series.

An important implication of the analytic Fredholm theorem is that the leading Laurent coefficients $\Upsilon_{-d},\ldots,\Upsilon_{-1}$ in (ref) are all of finite rank. In the representation theory to be developed, this has the effect of ensuring that we always obtain a cointegrating space with finite codimension (i.e., with finite dimensional orthogonal complement).

It will be convenient to introduce some additional notation. We will write $\Pi_0$ and $\Pi_1$ for the values taken by $\Phi(z)$ and its first derivative at $z=1$:

equation*[equation* omitted — 58 chars of source]

We also define the linear spaces

equation*[equation* omitted — 83 chars of source]

Our compactness condition on the $\Phi_j$'s ensures that $\alpha_1$ and $\beta_1$ are closed linear spaces with equal and finite codimension.

We have yet to give a formal definition of the I($d$) property. For our purposes, it is sufficient to define the I(0) property for standard linear processes. We will need to consider standard linear processes in $\mathcal H$ and in $\mathbb C$, with innovations in $\mathcal H$, so give the following definition of a standard linear process in an arbitrary separable complex Hilbert space $\mathcal H'$.

definitionA sequence $(W_t,t\geq t_0)$ in $L^2_{\mathcal H'}$ is called a standard linear process in $\mathcal H'$ if there is another separable complex Hilbert space $\mathcal H$ such that we may write \begin{equation} W_t=\sum_{k=0}^\infty A_k(\varepsilon_{t-k}),\quad t\geq t_0, \end{equation} where $(A_k,k\geq0)$ is a norm-summable sequence in $\mathcal{L}_{\mathcal H, \mathcal H'}$, and $(\varepsilon_t,t\in\mathbb Z)$ is an iid sequence in $L^2_{\mathcal H}$ with nonzero covariance operator $\Sigma\in\mathcal L_{\mathcal H}$.

remarkIf we were to require that the two Hilbert spaces $\mathcal H$ and $\mathcal H'$ are the same, and that $A_0=\mathrm{I}$, then our definition of a standard linear process in $\mathcal H$ would be the same as that of B00. Our more general definition is needed because if $(W_t,t\geq t_0)$ is a standard linear process in $\mathcal H$ with innovations in $\mathcal H$, then for any $x\in\mathcal H$ we may write \begin{equation*} \langle x,W_t\rangle=\sum_{k=0}^\infty \langle x,A_k(\varepsilon_{t-k})\rangle=\sum_{k=0}^\infty \hat{A}^x_k(\varepsilon_{t-k}),\quad t\geq t_0, \end{equation*} where the $\hat{A}^x_k$'s are given by \begin{equation*} \hat{A}^x_k(y)=\langle x,A_k(y)\rangle,\quad y\in\mathcal H, \end{equation*} and form a norm-summable sequence in $\mathcal{L}_{\mathcal H, \mathbb C}$. The sequence of inner products $(\langle x,W_t\rangle,t\geq t_0)$ is thus a standard linear process in $\mathbb C$ with innovations in $\mathcal H$.

We may now define the I($d$) property for sequences in $L^2_{\mathcal H'}$, with $\mathcal H'$ an arbitrary separable complex Hilbert space.

definitionWe say that a sequence $(W_t,t\geq t_0)$ in $L^2_{\mathcal H'}$ is I(0) if it is a standard linear process in $\mathcal H'$ admitting a representation (ref) in which $\Sigma$ is positive definite and the $A_k$'s satisfy $\sum_{k=0}^\infty A_k\neq0$ and $\sum_{k=0}^\infty k\Vert A_k\Vert_{\mathcal L_{\mathcal H,\mathcal H'}}<\infty$.

definitionWe say that a sequence $(W_t,t\geq t_0)$ in $L^2_{\mathcal H'}$ is I($d$) for $d\in\mathbb N$ if its $d$th difference is an I(0) standard linear process in $\mathcal H'$.

remarkThe summability condition on the norms of the coefficients $A_k$ in Definition (ref) is called 1-summability. It was used by PS92 to facilitate a version of the Beveridge-Nelson decomposition for a time series whose difference is I(0). In the results to be developed, all processes claimed to be I(0) in fact have coefficients decaying exponentially in norm, so 1-summability is easily satisfied.

Our first result provides an I(1) representation for autoregressive Hilbertian processes for which $\Phi(z)^{-1}$ has a simple pole at $z=1$. We will discuss the simple pole condition in more detail later in this section.

thmSuppose that Assumption (ref) is satisfied, and that $\Phi(z)^{-1}$ has a simple pole at $z=1$. Let $\Upsilon_{-1}$ denote the residue of $\Phi(z)^{-1}$ at $z=1$, let $\tilde{\Psi}(z) $ denote the analytic part of the Laurent series of $ \Phi(z)^{-1} $ around $z=1$, and set $ \tilde{\Psi}_k = \tilde{\Psi}^{(k)}(0)/k! $. A sequence $ (X_t, t \geq -p+1) $ in $L^2_\mathcal{H}$ satisfying the law of motion (ref) allows the following representation: for some $Z_0\in L^2_\mathcal{H} $ and all $t\geq1$ we have \begin{equation} X_t = Z_0 -\Upsilon_{-1}\left(\sum_{s=1}^t \varepsilon_{s}\right) + \nu_t . \end{equation} Here, $ (\nu_t,t\geq1)$ is a stationary sequence of random elements of $\mathcal{H}$ defined by the $L^2_\mathcal{H}$-convergent series $\nu_t=\sum_{k=0}^\infty \tilde{\Psi}_k (\varepsilon_{t-k}) $. Moreover, \begin{itemize} • The range of $\Upsilon_{-1}$ is equal to $\beta_1^\perp$ and has positive and finite dimension; • If $Z_0$ belongs to $\beta_1^\perp$, then for nonzero $x\in \mathcal{H}$ the sequence of inner products $(\langle x,X_t\rangle,t\geq1)$ is $\mathrm{I}(0)$ if $x\in\beta_1$, and is $\mathrm{I}(1)$ otherwise. \end{itemize}

remark\hyperref[mainthm]{Theorem \ref*{mainthm}} is similar to Theorem 4.1 of BSS17, but makes the connection to the analytic behavior of $\Phi(z)^{-1}$ explicit. The latter result is more generally applicable in one respect: compactness of the autoregressive operator is not assumed when $p=1$. The approach taken here relies on the analytic Fredholm theorem and therefore requires $\Phi(z)$ to be Fredholm, which may not be the case if the autoregressive operators are not compact.

comment\begin{remark} \hyperref[polethm]{Theorem \ref*{polethm}} and \hyperref[polethmp1]{Corollary \ref*{polethmp1}} provide several equivalent reformulations of the direct sum condition appearing in \hyperref[mainthm]{Theorem \ref*{mainthm}}, some of which may be simpler to verify in practice. \end{remark}

remarkThe residue $\Upsilon_{-1}$ appearing in \hyperref[mainthm]{Theorem \ref*{mainthm}} has finite rank by the analytic Fredholm theorem. The attractor space, which is the subspace of $\mathcal{H}$ in which the I(1) stochastic trend in the Beveridge-Nelson representation (ref) takes values, thus has finite dimension. We are therefore outside the framework considered by CHP16, in which the cointegrating space has finite dimension and the attractor space has finite codimension.

When can we expect the simple pole condition in Theorem (ref) to be satisfied? Our next result provides equivalent reformulations of this condition that may be easier to check in practice, and a formula for the residue $\Upsilon_{-1}$ in terms of $\alpha_1$, $\beta_1$ and $\Pi_1$. (Recall we defined $\Pi_1=\Phi^{(1)}(1)$.)

thmSuppose that Assumption (ref)(ii) holds. The following four conditions are equivalent. \begin{itemize} • $\Phi(z)^{-1}$ has a simple pole at $z=1$. • The operator $\Lambda_1:\beta_1^\perp\to\alpha_1^\perp$ obtained by restricting $\mathrm{P}_{\alpha_1^\perp}\Pi_1$ to $\beta_1^\perp$ is bijective. • $\mathcal{H}=\alpha_1\oplus\Pi_1\beta_1^\perp$. • $\mathcal{H}=\alpha_1+\Pi_1\beta_1^\perp$. \end{itemize} If $\Phi(z)^{-1}$ has a simple pole at $z=1$, then its residue at $z=1$ is $\Upsilon_{-1}=\Lambda_1^{-1}\mathrm{P}_{\alpha_1^\perp}$.

remarkThe closest results we have found to \hyperref[polethm]{Theorem \ref*{polethm}} in prior literature are those of S68 and H71, who worked in a more general Banach space setting. S68 established sufficient conditions for a simple pole, and H71 established the equivalence of conditions (1) and (3).

remarkBSS17 showed that condition (3) of Theorem (ref) is equivalent to the I(1) condition given by J91 in the finite dimensional case $\mathcal H=\mathbb C^n$. In this setting we may let $r<n$ be the rank of the $n\times n$ complex matrix $\Pi_0$, let $\alpha$ and $\beta$ be full-rank $n\times r$ complex matrices such that $\Pi_0=\alpha\beta^\prime$, and let $\alpha_\perp$ and $\beta_\perp$ be full-rank $n\times(n-r)$ complex matrices such that $\alpha^\prime\alpha_\perp=0$ and $\beta^\prime\beta_\perp=0$. The Johansen I(1) condition is satisfied when the $(n-r)\times(n-r)$ complex matrix $\alpha_\perp^\prime\Pi_1\beta_\perp$ is invertible. Examples 4.1--4.3 of BSS17 illustrate the reformulation of the Johansen I(1) condition as a direct sum decomposition.

remarkThe direct sum appearing in condition (3) of Theorem (ref) is not in general an orthogonal direct sum. FP18 showed that, when $\Phi(z)$ is noninvertible at $z=1$, condition (3) is equivalent to the following orthogonal direct sum decomposition of $\mathcal H$: \[\mathcal H=\beta_1\oplus\left(\ker\Lambda_1\mathrm{P}_{\beta_1^\perp}\right)^\perp. \] In their notation, this is $\mathcal H=\tau_0\oplus\tau_1$.

remarkOur assumption that the operators $\Phi_1,\ldots,\Phi_p$ are compact implies that $\Pi_0$ is Fredholm of index zero. If $\Pi_0$ were Fredholm but not of index zero then it would be impossible to satisfy condition (2). This is because bijectivity of $\Lambda_1$ requires its domain and codomain to have the same dimension. However, our proof that condition (4) implies condition (1) does not use the index-zero property, and remains valid if the compactness condition on $\Phi_1,\ldots,\Phi_p$ is weakened to require only that $\Phi(z)$ is Fredholm operator-valued.

comment\begin{remark} Condition (2) is the one that most closely resembles the condition used in the classical case $H=\mathbb C^n$ to establish I(1) representation theory. That condition, as given by S86,S91, is as follows. Let $A(z)$ be an $n\times n$ matrix depending holomorphically on $z$ in a neighborhood of $z_0$, nonsingular except at $z_0$, where it has rank $n-r$. Let $\alpha_\perp$ and $\beta_\perp$ be $n\times r$ matrices of full column rank such that $\alpha_\perp'A(z_0)=0$ and $A(z_0)\beta_\perp=0$. Then $A(z)^{-1}$ has a simple pole at $z_0$ if and only if $\alpha_\perp'A^{(1)}(z_0)\beta_\perp$ is invertible. To relate this condition to our condition (2), observe that since $\mathbb C^r$, $\ker A(z_0)$ and $\operatorname{coker} A(z_0)$ are isomorphic we may view $\alpha_\perp$ as a linear map from $\operatorname{coker} A(z_0)$ to $\mathbb C^n$ and $\beta_\perp$ as a linear map from $\ker A(z_0)$ to $\mathbb C^n$, and set $\alpha_\perp=\mathrm{id}_{\mathbb C^n}{\restriction_{\operatorname{coker} A(z_0)}}$ and $\beta_\perp=\mathrm{id}_{\mathbb C^n}{\restriction_{\ker A(z_0)}}$. Noting that $\alpha_\perp^\ast$ is the orthogonal projection operator $\mathrm{P}_{\operatorname{coker} A(z_0)}$ with codomain restricted to $\operatorname{coker} A(z_0)$, we find that $B_1=\alpha_\perp^\ast A^{(1)}(z_0)\beta_\perp$. Thus the bijectivity of $B_1$ asserted in our condition (2) is equivalent to the invertibility of $\alpha_\perp'A^{(1)}(z_0)\beta_\perp$ asserted in the classical condition. \end{remark}

In the special case where $p=1$, conditions (3) and (4) of \hyperref[polethm]{Theorem \ref*{polethm}} take on a particularly simple form, and another related equivalent condition becomes available. Moreover, the direct sum decomposition asserted by condition (3) serves to define an oblique projection that is the negative of the residue of our simple pole. The following corollary to \hyperref[polethm]{Theorem \ref*{polethm}} provides details.

corSuppose that Assumption (ref)(ii) holds and that $p=1$. The following four conditions are equivalent. \begin{itemize} • $\Phi(z)^{-1}$ has a simple pole at $z=1$. • $\mathcal{H}=\alpha_1\oplus\beta_1^\perp$. • $\mathcal{H}=\alpha_1+\beta_1^\perp$. • $\{0\}=\alpha_1\cap\beta_1^\perp$. \end{itemize} If $\Phi(z)^{-1}$ has a simple pole at $z=1$, then its residue at $z=1$ is the negative of the projection on $\beta_1^\perp$ along $\alpha_1$.

remarkThe oblique projection appearing in \hyperref[polethmp1]{Corollary \ref*{polethmp1}} is in fact the Riesz projection for the unit eigenvalue of $\Phi_1$. Said Riesz projection is defined (GGK90; M12) by the contour integral \begin{equation} P=\frac{1}{2\greektext p\mathrm{i}}\oint_\Gamma(z\mathrm{I}-\Phi_1)^{-1}\mathrm{d}z, \end{equation} where $\Gamma$ is a positively oriented smooth Jordan curve around one separating it from zero and from any other eigenvalues of $\Phi_1$, and where the integral of an $\mathcal L_\mathcal{H}$-valued function should be understood in the sense of Bochner. Let $\gamma:[0,1]\to\mathbb C$ be a smooth parametrization of $\Gamma$, and rewrite (ref) as \begin{equation} P=\frac{1}{2\greektext p\mathrm{i}}\int_0^1(\gamma(t)\mathrm{I}-\Phi_1)^{-1}\gamma'(t)\mathrm{d}t. \end{equation} The image of $\Gamma$ under the reciprocal transform $z\mapsto z^{-1}$, which we denote $\Gamma'$, is a positively oriented smooth Jordan curve around one separating it from any other poles of $\Phi(z)^{-1}$ and from zero. It admits the parametrization $t\mapsto1/\gamma(t)\eqqcolon\delta(t)$. A little calculus shows that $\gamma'(t)=-\delta'(t)/\delta(t)^2$, and so from (ref) we have \begin{equation} P=\frac{-1}{2\greektext p\mathrm{i}}\int_0^1\delta(t)^{-1}(\mathrm{I}-\delta(t)\Phi_1)^{-1}\delta'(t)\mathrm{d}t =\frac{-1}{2\greektext p\mathrm{i}}\oint_{\Gamma'} z^{-1}\Phi(z)^{-1}\mathrm{d}z. \end{equation} The residue theorem therefore tells us that $P$ is the negative of the residue of $z^{-1}\Phi(z)^{-1}$ at $z=1$, implying that the residue of $\Phi(z)^{-1}$ at $z=1$ is $-P$. It now follows from \hyperref[polethmp1]{Corollary \ref*{polethmp1}} that when the direct sum decomposition $\mathcal{H}=\alpha_1\oplus\beta_1^\perp$ is satisfied, the Riesz projection for the unit eigenvalue of $\Phi_1$ is the projection on $\beta_1^\perp$ along $\alpha_1$. This is a nonorthogonal projection except when $\alpha_1=\beta_1$, which would occur if, for instance, $\Phi_1$ is a normal operator.

remarkIt is apparent from our discussion in Remark (ref) that, under any of the equivalent I(1) conditions in Corollary (ref), the restriction of $\Pi_0$ to the range of the Riesz projection $P$ is zero. This is precisely the condition given by HP17 for an AR(1) process in $\mathcal H$ to be I(1).

We close this section with several examples of the use of Corollary (ref) for verifying that $\Phi(z)^{-1}$ has a simple pole at $z=1$.

exampleSuppose that $p=1$ and that $\Phi_1$ is self-adjoint. Then $\Pi_0$ is also self-adjoint, implying that $\alpha_1=\beta_1$. The direct sum decomposition $\mathcal{H} = \alpha_1 \oplus \beta_1^\perp$ appearing in Corollary (ref) is therefore satisfied, and is in this case an orthogonal direct sum decomposition.

exampleLet $(e_j , j \in \mathbb{N})$ be an orthonormal basis of $\mathcal{H}$. Suppose that $p=1$, and that $\Phi_1$ is given by \begin{equation*} \Phi_1(x) = \langle x, e_1\rangle (e_1+e_2) + \sum_{j=2}^\infty \lambda_j \langle x, e_j\rangle e_j,\quad x\in \mathcal{H}, \end{equation*} with $\lambda_j\in(0,1)$ for $j\geq2$ and $\lambda_j \rightarrow 0$ as $j \rightarrow \infty$. For any $x \in \mathcal{H}$ with representation $x= \sum_{j=1}^\infty c_j e_j$, $c_j=\langle x,e_j\rangle$, we have \begin{align} (\mathrm{I}-\Phi_1)(x) = (c_2(1-\lambda_2)-c_1)e_2 + \sum_{j=3}^\infty c_j(1-\lambda_j)e_j. \end{align} Since $\lambda_j\neq1$ for all $j\geq3$, it is clear that $e_j\notin\ker (\mathrm{I}-\Phi_1)$ for all $j\geq3$. Moreover, \begin{align*} (\mathrm{I}-\Phi_1)(c_1 e_1 +c_2 e_2) = (c_2 (1-\lambda_2) - c_1) e_2. \end{align*} It follows that \begin{align} \beta_1^\perp=\ker (\mathrm{I}-\Phi_1)=\{c_1e_1+c_2e_2:c_1=c_2(1-\lambda_2)\}. \end{align} Moreover, it may be deduced that $\alpha_1=\operatorname{cl}\operatorname{sp}\{e_j:j\geq2\}$, the closed linear span of $\{e_j:j\geq2\}$, as follows. Any $x\in\operatorname{cl}\operatorname{sp}\{e_j:j\geq2\}$ may be written as $x = \sum_{j=2}^\infty d_je_j$ for some square-summable sequence $ (d_j, j\geq2) $. We can always find another square-summable sequence $(c_j, j \in \mathbb{N})$ such that \begin{align} d_2=c_2(1-\lambda_2) - c_1 \quad and \quad d_j = c_j(1-\lambda_j),\quad j\geq3. \end{align} Then \begin{align*} (\mathrm{I}-\Phi_1)\left(\sum_{j=1}^\infty c_je_j\right)=(c_2(1-\lambda_2)-c_1)e_2 + \sum_{j=3}^\infty c_j(1-\lambda_j)e_j=\sum_{j=2}^\infty d_je_j=x, \end{align*} which shows that $x\in\operatorname{ran}(\mathrm{I}-\Phi_1)=\alpha_1$. Thus $\operatorname{cl}\operatorname{sp}\{e_j:j\geq2\}\subseteq\alpha_1 $. In addition, it is easily deduced that $\alpha_1 \subseteq \operatorname{cl}\operatorname{sp}\{e_j:j\geq2\} $ using (ref). Therefore, $\alpha_1 =\operatorname{cl}\operatorname{sp}\{e_j:j\geq2\}$. From (ref) we see that the only element of $\beta_1^\perp$ belonging to $\operatorname{cl}\operatorname{sp}\{e_j:j\geq2\}$ is zero. Thus the condition $\{0\}=\alpha_1\cap\beta_1^\perp$ appearing in \hyperref[polethmp1]{Corollary \ref*{polethmp1}} is satisfied.

exampleSuppose that in Example (ref) we instead defined $\Phi_1\in\mathcal L_\mathcal{H}$ by \begin{equation*} \Phi_1(x) = \langle x, e_1\rangle (e_1+e_2+e_3) + \langle x, e_2\rangle e_2 + \langle x, e_3\rangle e_3 + \sum_{j=4}^\infty \lambda_j \langle x, e_j\rangle e_j,\quad x\in \mathcal{H}, \end{equation*} with $\lambda_j\in(0,1)$ for $j\geq4$ and $\lambda_j \rightarrow 0$ as $j \rightarrow \infty$. For any $x \in \mathcal{H}$ with representation $x= \sum_{j=1}^\infty c_j e_j$, $c_j=\langle x,e_j\rangle$, we now have \begin{align*} (\mathrm{I}-\Phi_1)(x) =& -c_1e_2 -c_1e_3+ \sum_{j=4}^\infty c_j(1-\lambda_j)e_j. \end{align*} Since $\lambda_j\neq1$ for all $j\geq4$, it is clear that $e_j \notin \ker (\mathrm{I}-\Phi_1) $ for all $j\geq4$. Moreover, \begin{align} (\mathrm{I}-\Phi_1)(c_1 e_1 +c_2 e_2 +c_3 e_3) = -c_1 e_2 - c_1 e_3. \end{align} It follows that $\beta_1^\perp=\ker (\mathrm{I}-\Phi_1)=\operatorname{sp} \{e_2, e_3\}$. Further, arguments similar to those in Example (ref) can be used to show that \begin{align*} \alpha_1 = \operatorname{cl}\operatorname{sp}\{e_2+e_3, e_4, e_5, \ldots\}. \end{align*} It follows that $\alpha_1\cap\beta_1^\perp=\operatorname{sp}\{e_2+e_3\}$. Thus the condition $\{0\}=\alpha_1\cap\beta_1^\perp$ appearing in \hyperref[polethmp1]{Corollary \ref*{polethmp1}} is violated.

I(2) autoregressive Hilbertian processes

In this section we state our results for I(2) autoregressive processes. We continue to assume that the autoregressive coefficients and innovations of such a process satisfy Assumption (ref). It will be convenient to introduce some additional notation. We define the linear spaces

equation[equation omitted — 121 chars of source]

The role played by $\alpha_2$ and $\beta_2$ in our I(2) results will be analogous to the role played by $\alpha_1$ and $\beta_1$ in our I(1) results. The notation $\Pi_1^\ast$ refers to the adjoint operator to $\Pi_1$.

remarkThe linear spaces $\alpha_2^\perp$ and $\beta_2^\perp$ have equal and finite dimension and are, respectively, the cokernel and kernel of the operator $\Lambda_1$ appearing in Theorem (ref). To see why, observe that $\alpha_2$ satisfies \begin{equation*} \alpha_2^\perp=(\alpha_1+\mathrm{P}_{\alpha_1^\perp}\Pi_1\beta_1^\perp)^\perp=(\alpha_1+\operatorname{ran}\Lambda_1)^\perp=\ker\Lambda_1^\ast, \end{equation*} and similarly $\beta_2$ satisfies \begin{equation*} \beta_2^\perp=(\beta_1+\mathrm{P}_{\beta_1^\perp}\Pi_1^\ast\alpha_1^\perp)^\perp=(\beta_1+\operatorname{ran}\Lambda_1^\ast)^\perp=\ker\Lambda_1. \end{equation*} The operator $\Lambda_1$ is invertible under any of the equivalent I(1) conditions given in Theorem (ref), in which case we must have $\alpha_2=\beta_2=\mathcal H$. In this section we are interested in the case where our I(1) conditions fail. This occurs when $\alpha_2^\perp$ and $\beta_2^\perp$ have positive dimension.

remarkSimpler expressions for $\alpha_2$ and $\beta_2$ become available when $p=1$. In this case we have $\Pi_1=\Pi_0-\mathrm{I}$, from which it follows easily that $\Pi_1\beta_1^\perp=\beta_1^\perp$ and $\Pi_1^\ast\alpha_1^\perp=\alpha_1^\perp$. We may therefore write \begin{equation*} \alpha_2=\alpha_1 + \beta_1^\perp,\quad \beta_2=\beta_1+\alpha_1^\perp. \end{equation*}

Our first result in this section provides an I(2) analogue to Theorem (ref) in Section (ref). It establishes an I(2) representation for autoregressive Hilbertian processes for which $\Phi(z)^{-1}$ has a pole of second order at $z=1$. We will discuss the pole condition in more detail later in this section.

thmSuppose that Assumption (ref) is satisfied, and that $\Phi(z)^{-1}$ has a pole of second order at $z=1$. Let $\Upsilon_{-2}$ and $\Upsilon_{-1}$ denote the coefficients in the principal part of the Laurent series of $ \Phi(z)^{-1} $ around $z=1$, let $\tilde{\Psi}(z) $ denote the analytic part of the Laurent series of $ \Phi(z)^{-1} $ around $z=1$, and set $ \tilde{\Psi}_k = \tilde{\Psi}^{(k)}(0)/k! $. A sequence $ (X_t, t \geq -p+1) $ in $L^2_\mathcal{H}$ satisfying the law of motion (ref) allows the following representation: for some $Z_0,Z_1\in L^2_\mathcal{H} $ and all $t\geq1$ we have \begin{equation} X_t = Z_0+tZ_1 + \Upsilon_{-2}\left(\sum_{s=1}^t\sum_{r=1}^s \varepsilon_{r}\right)-\Upsilon_{-1}\left(\sum_{s=1}^t\varepsilon_{s}\right) + \nu_t . \end{equation} Here, $ (\nu_t,t\geq1)$ is a stationary sequence of random elements of $\mathcal{H}$ defined by the $L^2_\mathcal{H}$-convergent series $\nu_t = \sum_{k=0}^\infty \tilde{\Psi}_k (\varepsilon_{t-k}) $. Moreover, \begin{itemize} • The range of $\Upsilon_{-2}$ is equal to $\beta_2^\perp$ and has positive and finite dimension; • If $Z_0$ and $Z_1$ belong to $\beta_2^\perp$, then for nonzero $x\in \mathcal{H}$ the sequence of inner products $(\langle x,X_t\rangle,t\geq1)$ is $\mathrm{I}(0)$ or $\mathrm{I}(1)$ if $x\in\beta_2$, and is $\mathrm{I}(2)$ otherwise. \end{itemize} \begin{comment} If we suppose further that $p=1$, then \begin{itemize} • If $Z_0$ and $Z_1$ belong to $\Pi_0^{-1}(\alpha_1\cap\beta_1^\perp)+\beta_1^\perp$ with probability one, then for nonzero $x\in \mathcal{H}$ the sequence of inner products $(\langle X_t,x\rangle,t\geq1)$ is $\mathrm{I}(0)$ if and only if $x\in(\Pi_0^{-1}(\alpha_1\cap\beta_1^\perp))^\perp\cap\beta_1$, and is $\mathrm{I}(1)$ or $\mathrm{I}(2)$ otherwise; • The linear spaces $\Pi_0^{-1}(\alpha_1\cap\beta_1^\perp)+\beta_1^\perp$, $\operatorname{ran}\Upsilon_{-1}$ and $\operatorname{ran}\Upsilon_{-2}+\operatorname{ran}\Upsilon_{-1}$ are equal to one another and have positive and finite dimension. \end{itemize} \end{comment}

In view of claim (2) in Theorem (ref) we may refer to the linear subspace $\beta_2$ as the cointegrating space. For $x\in\beta_2$, the sequence of inner products $(\langle x,\Delta X_t\rangle,t\geq1)$ may be $\mathrm{I}(0)$ or $\mathrm{I}(1)$ (ignoring the effect of the deterministic components $Z_0$ and $Z_1$). Polynomial cointegration may also occur. This will be discussed more fully in Remark (ref) below.

Our next result provides an I(2) analogue to Theorem (ref) in Section (ref). It establishes necessary and sufficient conditions for $\Phi(z)^{-1}$ to have a pole of second order at $z=1$, as assumed in Theorem (ref). It also establishes formulas for the Laurent coefficients $\Upsilon_{-2}$ and $\Upsilon_{-1}$ appearing in Theorem (ref). To state our result it will be convenient to introduce some additional notation. Similar to $\Pi_0$ and $\Pi_1$, we define

equation*[equation* omitted — 133 chars of source]

Here and elsewhere, a plus superscript is used to denote the Moore-Penrose inverse of a continuous linear operator between Hilbert spaces with closed range BG03.

thmSuppose that Assumption (ref)(ii) holds, and that $\Phi(z)^{-1}$ does not have a simple pole at $z=1$. Then the following four conditions are equivalent. \begin{itemize} • $\Phi(z)^{-1}$ has a pole of second order at $z=1$. • The operator $\Lambda_2: \beta_2^\perp \to \alpha_2^\perp$ obtained by restricting $\mathrm{P}_{\alpha_2^\perp}\Pi_2$ to $\beta_2^\perp$ is bijective. • $\mathcal{H}=\alpha_2 \oplus \Pi_2 \beta_2^\perp$. • $\mathcal{H}=\alpha_2+ \Pi_2 \beta_2^\perp$. \end{itemize} If $\Phi(z)^{-1}$ has a pole of second order at $z=1$, then the coefficients of $(z-1)^{-2}$ and $(z-1)^{-1}$ in the Laurent series of $ \Phi(z)^{-1} $ around $z=1$ are given by \begin{equation} \Upsilon_{-2}=\Lambda_2^{-1}\mathrm{P}_{\alpha_2^\perp} \end{equation} and \begin{align} \Upsilon_{-1}&=\Lambda_1^+ \mathrm{P}_{\alpha_1^\perp}-\left[\Lambda_1^+ \mathrm{P}_{\alpha_1^\perp}\Pi_2+\Pi_0^+ \Pi_1\right]\Upsilon_{-2}-\Upsilon_{-2}\left[\Pi_2\Lambda_1^+ \mathrm{P}_{\alpha_1^\perp}+\Pi_1\Pi_0^+\right]\notag\\ &\quad+\Upsilon_{-2}\left[\Pi_1\Pi_0^+\Pi_2+\Pi_2\Pi_0^+\Pi_1+\Pi_2\Lambda_1^+\mathrm{P}_{\alpha_1^\perp}\Pi_2-\Pi_3\right]\Upsilon_{-2}, \end{align} respectively. \begin{comment} If $\Phi(z)^{-1}$ has a pole of second order at $z=1$, then the coefficient of $(z-1)^{-2}$ in the Laurent series of $ \Phi(z)^{-1} $ around $z=1$ is the operator $\Upsilon_{-2}\in\mathcal L_\mathcal{H}$ given by \begin{equation} \Upsilon_{-2}(x)=\Lambda_2^{-1}\mathrm{P}_{\alpha_2^\perp}(x),\quad x\in \mathcal{H}, \end{equation} and the coefficient of $(z-1)^{-1}$ in the Laurent series of $A(z)^{-1}$ around $z=1$ has the representation \begin{align*} &\Upsilon_{-1} = \Upsilon_{-1} \mathrm{P}_{\alpha_1} + \Upsilon_{-1}\mathrm P_{\alpha_1^\perp\cap\alpha_2} + \Upsilon_{-1} \mathrm P_{\alpha_2^\perp}, \end{align*} where \begin{align} \Upsilon_{-1} \mathrm{P}_{\alpha_1} &= - \Upsilon_{-2} \Pi_1 \Pi_0^+, \\ \Upsilon_{-1}\mathrm P_{\alpha_1^\perp\cap\alpha_2} &= (\mathrm{I} - \Upsilon_{-2} \Pi_2)\Lambda_1^+ \mathrm{P}_{\alpha_1^\perp},\\ \Upsilon_{-1} \mathrm P_{\alpha_2^\perp}&= \Upsilon_{-2} \left[ \Pi_1 \Pi_0^+ \Pi_2 + \Pi_2\Pi_0^+ \Pi_1 - \Pi_3 \right]\Upsilon_{-2} \\ &\quad - \left[\Pi_0^+ \Pi_1+(\mathrm{I} - \Upsilon_{-2} \Pi_2)\Lambda_1^+ \mathrm{P}_{\alpha_1^\perp} \Pi_2\right]\Upsilon_{-2}. \notag \end{align} \end{comment}

remarkThe four equivalent conditions in Theorem (ref) are equivalent to the I(2) condition given by J92 in the finite dimensional case $\mathcal H=\mathbb C^n$. Continuing with the notation of Remark (ref), suppose that the Johansen I(1) condition fails, and let $\varphi$ and $\eta$ be full-rank $(n-r)\times s$ ($s<n-r$) complex matrices such that $(\alpha_\perp^\prime\alpha_\perp)^{-1}\alpha_\perp^\prime\Pi_1\beta_\perp(\beta_\perp^\prime\beta_\perp)^{-1}=\varphi\eta^\prime$. Let $\tilde{\alpha}_\perp$ and $\tilde{\beta}_\perp$ be full-rank $n\times(n-r-s)$ complex matrices whose columns are orthogonal to those of $\tilde{\alpha}\coloneqq(\alpha,\alpha_\perp\varphi)$ and $\tilde{\beta}\coloneqq(\beta,\beta_\perp\eta)$ respectively. Note that the column spaces of $\tilde{\alpha}$ and $\tilde{\beta}$ are, respectively, the linear spaces $\alpha_2$ and $\beta_2$ defined in (ref) above. The Johansen I(2) condition is satisfied when the $(n-r-s)\times(n-r-s)$ complex matrix $\tilde{\alpha}_\perp^\prime\Pi_2\tilde{\beta}_\perp$ is invertible. Lemma 4.1 of BSS17 implies that $\tilde{\alpha}_\perp^\prime\Pi_2\tilde{\beta}_\perp$ is invertible if and only if $\mathcal H$ is the direct sum of the null space of $\tilde{\alpha}_\perp^\prime$ and the column space of $\Pi_2\tilde{\beta}_\perp$. The former space is $\alpha_2$ and the latter space is $\Pi_2\beta_2^\perp$, so the Johansen I(2) condition is equivalent to condition (3) in Theorem (ref).

remarkFormulas (ref) and (ref) in Theorem (ref) correspond to formulas (12) and (13) of J09, given for the finite dimensional case $\mathcal H=\mathbb C^n$. The objects $C_1$, $C_2$, $\bar{\beta}_1\bar{\alpha}_1^\prime$, $\bar{\beta}\bar{\alpha}'$, $\dot{\Pi}$, $\theta$ and $(1/6)\dddot{\Pi}-\dot{\Pi}\bar{\beta}\bar{\alpha}'\dot{\Pi}\bar{\beta}\bar{\alpha}'\dot{\Pi}$ in Johansen's notation correspond respectively to $-\Upsilon_{-1}$, $\Upsilon_{-2}$, $\Lambda_1^+\mathrm{P}_{\alpha_1^\perp}$, $-\Pi_0^+$, $\Pi_1$, $\Pi_2$ and $\Pi_3$ in our notation.

remarkThe direct sum appearing in condition (3) of Theorem (ref) is not in general an orthogonal direct sum. Extending results of J92 from the finite dimensional case $\mathcal H=\mathbb C^n$ to a more general Hilbert space setting, FP18 showed that, when $\Phi(z)$ is noninvertible at $z=1$ and the I(1) condition fails, an equivalent necessary and sufficient condition for a pole of second order is the following tripartite orthogonal direct sum decomposition of $\mathcal H$: \begin{equation*} \mathcal H=\beta_1\oplus\left(\ker\Lambda_1\mathrm{P}_{\beta_1^\perp}\right)^\perp\oplus\left(\ker\Lambda_2\mathrm{P}_{\beta_2^\perp}\right)^\perp. \end{equation*} In their notation, this is $\mathcal H=\tau_0\oplus\tau_1\oplus\tau_2$. Moreover, we have the following bipartite orthogonal direct sum decomposition of the cointegrating space $\beta_2$: \begin{equation*} \beta_2=\beta_1\oplus\left(\ker\Lambda_1\mathrm{P}_{\beta_1^\perp}\right)^\perp, \end{equation*} or $\beta_2=\tau_0\oplus\tau_1$. This decomposition is informative because the two subspaces decomposing the cointegrating space correspond to different kinds of cointegrating behavior. Suppose that $Z_0$ and $Z_1$ belong to $\beta_2^\perp$, so that we may ignore the effect of deterministic components. For $x\in\beta_2$ not belonging to $\beta_1$, the sequence $(\langle x,X_t\rangle,t\geq1)$ is $\mathrm{I}(1)$. For nonzero $x\in\beta_1$, the sequence $(\langle x,X_t\rangle,t\geq1)$ may be $\mathrm{I}(0)$ or $\mathrm{I}(1)$, but the sequence $(\langle x,X_t\rangle-\langle x,\Pi_0^+\Pi_1\Delta X_t\rangle,t\geq1)$ is always $\mathrm{I}(0)$. The case where the inner product sequences $(\langle x,X_t\rangle,t\geq1)$ and $(\langle x,\Pi_0^+\Pi_1\Delta X_t\rangle,t\geq1)$ are $\mathrm{I}(1)$ and cointegrated is called polynomial cointegration or multicointegration Y87,GL89,GL90,EJ99,PK19.

comment\begin{remark} It is apparent from the description of multicointegration given in Remark (ref) that the subspace $(\Pi_0^+\Pi_1)^\ast\beta_1$ plays an important role: given any nonzero $x\in\beta_1$, there exists a nonzero $y\in(\Pi_0^+\Pi_1)^\ast\beta_1$ such that the sequence $(\langle x,X_t\rangle-\langle y,\Delta X_t\rangle,t\geq1)$ is I(0) (ignoring deterministic components). Similarly, given any nonzero $y\in(\Pi_0^+\Pi_1)^\ast\beta_1$, there exists a nonzero $x\in\beta_1$ such that the sequence $(\langle x,X_t\rangle-\langle y,\Delta X_t\rangle,t\geq1)$ is I(0). A simpler expression for the subspace $(\Pi_0^+\Pi_1)^\ast\beta_1$ is available: noting that $\Pi_0^\ast$ has range $\beta_1$ and corange $\alpha_1$, we have \begin{equation*} (\Pi_0^+\Pi_1)^\ast\beta_1=\Pi_1^\ast(\Pi_0^+)^\ast\beta_1=\Pi_1^\ast(\Pi_0^\ast)^+\beta_1=\Pi_1^\ast\alpha_1. \end{equation*} \end{remark}

remarkFrom our discussion of the $\mathrm{I}(2)$ case in Remark (ref), it is apparent that (ignoring deterministic components) the sequence $(\langle x,X_t\rangle,t\geq1)$ is $\mathrm{I}(0)$ for precisely those nonzero $x\in\beta_1$ for which $(\langle x,\Pi_0^+\Pi_1\Delta X_t\rangle,t\geq1)$ is stationary. This is the collection of all nonzero $x\in\beta_1$ such that $(\Pi_0^+\Pi_1)^\ast(x)$ belongs to $\beta_2$, the cointegrating space. Therefore, $(\langle x,X_t\rangle,t\geq1)$ is $\mathrm{I}(0)$ if and only if $x$ is a nonzero element of the linear space \begin{equation*} \zeta=\beta_1\cap(\Pi_0^+\Pi_1\beta_2^\perp)^\perp. \end{equation*}

remarkWhen $p=1$, the linear space $\Pi_2\beta_2^\perp$ appearing in conditions (3) and (4) of Theorem (ref) is equal to $(\mathrm{I}-\Pi_0^+)(\alpha_1\cap\beta_1^\perp)$. To see why, observe that when $p=1$ we have \begin{equation*} \Pi_2=-\Pi_1\Pi_0^+\Pi_1=-(\mathrm{I}-\Pi_0)\Pi_0^+(\mathrm{I}-\Pi_0)=-\Pi_0^++\Pi_0^+\Pi_0+\Pi_0\Pi_0^+-\Pi_0\Pi_0^+\Pi_0. \end{equation*} From Remark (ref) we know that $\beta_2^\perp=\alpha_1\cap\beta_1^\perp$ when $p=1$. Therefore, since $\Pi_0$ is zero on $\beta_1^\perp$, we have \begin{equation*} \Pi_2\beta_2^\perp=(\Pi_0\Pi_0^+-\Pi_0^+)(\alpha_1\cap\beta_1^\perp). \end{equation*} From the basic properties of Moore-Penrose inverses, we know that $\Pi_0\Pi_0^+$ is orthogonal projection on $\alpha_1$, the range of $\Pi_0$. The restriction of this projection to $\alpha_1\cap\beta_1^\perp$ coincides with the identity, so our claim is established.

remarkWhen $p=1$, the linear space $\zeta$ appearing in Remark (ref) is equal to $\beta_1\cap(\Pi_0^+(\alpha_1\cap\beta_1^\perp))^\perp$. To see why, observe that when $p=1$ we have $\beta_2^\perp=\alpha_1\cap\beta_1^\perp$ (shown in Remark (ref)) and $\Pi_0^+\Pi_1=\Pi_0^+\Pi_0-\Pi_0^+$. From the basic properties of Moore-Penrose inverses, we know that $\Pi_0^+\Pi_0$ is orthogonal projection on $\beta_1$, the corange of $\Pi_0$. Thus $\Pi_0^+\Pi_0$ is zero on $\alpha_1\cap\beta_1^\perp$, and we have $\Pi_0^+\Pi_1\beta_2^\perp=\Pi_0^+(\alpha_1\cap\beta_1^\perp)$, establishing our claim.

commentDefine $\zeta$ to be the closed linear subspace of $\mathcal H$ whose orthogonal complement is given by \begin{align} \zeta^\perp&=\left[\Lambda_1^+-\left(\Lambda_1^+\mathrm{P}_{\alpha_1^\perp}\Pi_2+\Pi_0^+\Pi_1\right)\Lambda_2^{-1}\mathrm{P}_{\alpha_2^\perp}\right]\alpha_1^\perp+\beta_2^\perp. \end{align} \begin{thm} Suppose that the assumptions of Theorem (ref) are satisfied, so that the sequence $ (X_t, t \geq -p+1) $ in $L^2_\mathcal{H}$ allows the representation (ref). Then \begin{itemize} • The sum of the ranges of $\Upsilon_{-2}$ and $\Upsilon_{-1}$ is equal to $\zeta^\perp$ and has positive and finite dimension; • If $Z_0$ and $Z_1$ belong to $\zeta^\perp$, then for nonzero $x\in \mathcal{H}$ the sequence of inner products $(\langle x,X_t\rangle,t\geq1)$ is $\mathrm{I}(0)$ if and only if $x\in\zeta$, and is $\mathrm{I}(1)$ or $\mathrm{I}(2)$ otherwise. \end{itemize} If we suppose further that $p=1$, then the range of $\Upsilon_{-1}$ is equal to $\zeta^\perp$, and we have $\zeta^\perp=\Pi_0^+(\alpha_1\cap\beta_1^\perp)+\beta_1^\perp$. \end{thm}

comment\begin{thm} Suppose that Assumption (ref)(ii) holds, that $\Phi(z)^{-1}$ has a pole of second order at $z=1$, and that $p=1$. Then $\zeta^\perp=\beta_1^\perp+\Pi_0^+\beta_2^\perp$. \end{thm}

remarkAs discussed in Remark (ref), when $p=1$, the negative of the residue $\Upsilon_{-1}$ is a projection, and is called the Riesz projection for the unit eigenvalue of $\Phi_1$. The space on which $-\Upsilon_{-1}$ projects is called the generalized eigenspace for the unit eigenvalue, and its dimension is called the algebraic multiplicity of the unit eigenvalue. Contained within the generalized eigenspace is the usual eigenspace $\beta_1^\perp$, whose dimension is called the geometric multiplicity of the unit eigenvalue GGK90. The generalized eigenspace is in fact the orthogonal complement to $\zeta$, the subspace of the cointegrating space $\beta_2$ yielding $\mathrm{I}(0)$ inner products, as defined in Remark (ref). To see why, observe that $\zeta^\perp$ is the subspace occupied by the $\mathrm{I}(1)$ and $\mathrm{I}(2)$ trends in the representation (ref), which is the sum of the ranges of $\Upsilon_{-2}$ and $\Upsilon_{-1}$. But from formula (ref) in Theorem (ref), we know that the range of $\Upsilon_{-2}$ is $\beta_2^\perp$, which is contained in the usual eigenspace $\beta_1^\perp$, and therefore contained in the generalized eigenspace, which is the range of $\Upsilon_{-1}$. Therefore $\zeta^\perp$ is the generalized eigenspace. In view of Remark (ref), we have \begin{equation} \zeta^\perp=\beta_1^\perp+\Pi_0^+(\alpha_1\cap\beta_1^\perp). \end{equation} From Corollary (ref) we know that for $p=1$ the $\mathrm{I}(1)$ condition fails precisely when $\alpha_1\cap\beta_1^\perp\neq\{0\}$. Since the Moore-Penrose inverse $\Pi_0^+$ defines a bijection from $\alpha_1$ to $\beta_1$, and $\beta_1^\perp$ is finite dimensional, we deduce that when $p=1$ and the $\mathrm{I}(1)$ condition fails we must have $\dim\beta_1^\perp<\dim\zeta^\perp.$ Thus we see that when the $\mathrm{I}(1)$ condition fails, the algebraic multiplicity of the unit eigenvalue exceeds its geometric multiplicity. This contrasts with the situation when the $\mathrm{I}(1)$ condition is satisfied, where, as is apparent from our discussion in Remark (ref), the algebraic and geometric multiplicities of the unit eigenvalue are equal. The fact that the equality of geometric and algebraic multiplicities of the unit eigenvalue implies an $\mathrm{I}(1)$ representation was observed by J96 in the finite dimensional case $\mathcal H=\mathbb C^n$.

remarkHP17 provide the following condition for an AR(1) process in $\mathcal H$ to be $\mathrm{I}(2)$: the restriction of $\Pi_0$ to the generalized eigenspace $\zeta^\perp$ must be nilpotent of degree two. We can verify this condition using the expression we obtained for $\zeta^\perp$ in (ref). Based on this expression, any element of $\zeta^\perp$ may be written as $x+\Pi_0^+(y)$, where $x\in\beta_1^\perp$ and $y\in\alpha_1\cap\beta_1^\perp$. Observe that \begin{equation*} \Pi_0(x+\Pi_0^+(y))=\Pi_0\Pi_0^+(y)=y\quadand\quad\Pi_0^2(x+\Pi_0^+(y))=\Pi_0(y)=0, \end{equation*} since $\Pi_0\Pi_0^+$ is orthogonal projection on $\alpha_1$. Thus the restriction of $\Pi_0$ to $\zeta^\perp$ is zero if and only if $\alpha_1\cap\beta_1^\perp=\{0\}$, which is one of our necessary and sufficient conditions for a simple pole from Corollary (ref). Assuming that this condition fails, and that our pole is of second order, we find that the restriction of $\Pi_0$ to $\zeta^\perp$ is nilpotent of degree two.

figure[figure omitted — 1,545 chars of source]

To illustrate the preceding results on $\mathrm{I}(2)$ representations we discuss three examples. The first two examples correspond to Examples 4.2--4.3 of J96 and BSS17, with $\gamma=2$ in the latter example so that the solution is I(2). These examples are finite (two) dimensional.

exampleConsider the first-order autoregressive law of motion $X_t=\Phi_1(X_{t-1})+\varepsilon_t$ in $\mathcal H=\mathbb C^2$ with autoregressive coefficient matrix \begin{equation*} \Phi_1=\left[\begin{array}{cc}1&0\\1&1\end{array}\right]. \end{equation*} The only eigenvalue of $\Phi_1$ is one. With a little algebra we find that \begin{equation*} \alpha_1=\beta_1^\perp=\alpha_2=\beta_2^\perp=\mathrm{sp}\left[\begin{array}{c}0\\1\end{array}\right],\quad\Pi_2\beta_2^\perp=\mathrm{sp}\left[\begin{array}{c}1\\1\end{array}\right]. \end{equation*} We depict these subspaces of $\mathbb C^2$ in Figure (ref)(a), displaying them as subspaces of $\mathbb R^2$ since $\Phi_1$ has real elements. (We do the same for Example (ref) in Figure (ref)(b).) It is apparent that the (version for $p=1$ of the) I(1) condition $\mathbb C^2=\alpha_1\oplus\beta_1^\perp$ is not satisfied, and that the I(2) condition $\mathbb C^2=\alpha_2\oplus\Pi_2\beta_2^\perp$ is satisfied, so our autoregressive law of motion generates an I(2) process. It is easily verified that $\zeta=\{0\}$, so in this case it follows from Theorem (ref) that (ignoring deterministic components) the inner product sequence $(\langle x,X_t\rangle,t\geq1)$ is I(1) for nonzero $x\in\beta_2$ and I(2) for nonzero $x\notin\beta_2$.

exampleConsider the second-order autoregressive law of motion $X_t=\Phi_1(X_{t-1})+\Phi_2(X_{t-2})+\varepsilon_t$ in $\mathcal H=\mathbb C^2$ with autoregressive coefficient matrices \begin{equation*} \Phi_1=\left[\begin{array}{cc}\frac{5}{4}&\frac{7}{4}\\-\frac{1}{4}&\frac{5}{4}\end{array}\right],\quad\Phi_2=\left[\begin{array}{cc}0&-2\\0&0\end{array}\right]. \end{equation*} The points of noninvertibility of $\Phi(z)$ are $z=1$ and $z=2$. It is straightforward to establish that \begin{equation*} \alpha_1=\beta_1=\Pi_1\beta_1^\perp=\alpha_2=\beta_2=\mathrm{sp}\left[\begin{array}{c}1\\-1\end{array}\right],\quad\Pi_2\beta_2^\perp=\mathrm{sp}\left[\begin{array}{c}5\\-3\end{array}\right]. \end{equation*} We depict these subspaces in Figure (ref)(b). It is apparent that the I(1) condition $\mathbb C^2=\alpha_1\oplus\Pi_1\beta_1^\perp$ is not satisfied, and that the I(2) condition $\mathbb C^2=\alpha_2\oplus\Pi_2\beta_2^\perp$ is satisfied, so our autoregressive law of motion generates an I(2) process. As in Example (ref), we have $\zeta=\{0\}$, so that (ignoring deterministic components) $(\langle x,X_t\rangle,t\geq1)$ is I(1) for nonzero $x\in\beta_2$ and I(2) for nonzero $x\notin\beta_2$.

Our final example is infinite dimensional, and builds on Example (ref) above.

exampleConsider the setting of Example (ref). We can use Theorem (ref) to determine whether we have a pole of second order. The Moore-Penrose inverse $\Pi_0^+$ satisfies $\Pi_0^+\Pi_0=\mathrm{P}_{\beta_1}$. Applying $\Pi_0^+$ to both sides of the equality $(\mathrm{I}-\Phi_1)(-e_1)=e_2+e_3$ reveals that $\mathrm{P}_{\beta_1}(-e_1)=\Pi_0^+(e_2+e_3)$, which simplifies to $\Pi_0^+(e_2+e_3)=-e_1$ since $\beta_1=\operatorname{sp}\{e_2,e_3\}^\perp$. In view of Remark (ref), it follows that \begin{equation*} \Pi_2\beta_2^\perp=(\mathrm{I}-\Pi_0^+)(\alpha_1\cap\beta_1^\perp)=(\mathrm{I}-\Pi_0^+)\operatorname{sp}\{e_2+e_3\}=\operatorname{sp}\{e_1+e_2+e_3\}. \end{equation*} In view of Remark (ref), we have \begin{equation*} \alpha_2=\alpha_1+\beta_1^\perp=\operatorname{cl}\operatorname{sp}\{e_2+e_3,e_4,e_5,\ldots\}+\operatorname{sp}\{e_2,e_3\}=\operatorname{cl}\operatorname{sp}\{e_2,e_3,e_4,\ldots\}. \end{equation*} Thus we see that $\mathcal{H}$ is the sum of the linear subspaces $\alpha_2$ and $\Pi_2\beta_2^\perp$, and deduce from Theorem (ref) that $\Phi(z)^{-1}$ has a pole of second order at $z=1$. The associated cointegrating space is \begin{equation*} \beta_2=\beta_1+\alpha_1^\perp=\operatorname{sp}\{e_2,e_3\}^\perp+\operatorname{cl}\operatorname{sp}\{e_2+e_3,e_4,e_5,\ldots\}^\perp=\operatorname{cl}\operatorname{sp}\{e_1,e_2-e_3,e_4,\ldots\}. \end{equation*} The $\mathrm{I}(2)$ stochastic trend takes values in the orthogonal complement to this space, which is \begin{equation*} \beta_2^\perp=\operatorname{sp}\{e_2+e_3\}. \end{equation*} In view of Remark (ref), the $\mathrm{I}(1)$ stochastic trend takes values in the larger space \begin{equation*} \zeta^\perp=\beta_1^\perp+\Pi_0^+(\alpha_1\cap\beta_1^\perp)=\operatorname{sp}\{e_2,e_3\}+\operatorname{sp}\{e_1\}=\operatorname{sp}\{e_1,e_2,e_3\}. \end{equation*} It follows that (ignoring deterministic components) for nonzero $x\in\mathcal H$ the inner product sequence $(\langle x,X_t\rangle,t\geq1)$ is \begin{align*} I(0) for &x\in\operatorname{cl}\operatorname{sp}\{e_4,e_5,\ldots\},\\ I(1) for &x\in\operatorname{cl}\operatorname{sp}\{e_1,e_2-e_3,e_4,\ldots\}\setminus\operatorname{cl}\operatorname{sp}\{e_4,e_5,\ldots\},\\ I(2) for &x\notin\operatorname{cl}\operatorname{sp}\{e_1,e_2-e_3,e_4,\ldots\}. \end{align*}

Concluding remarks

The results established in this paper extend long-known results on I(1) and I(2) representations for autoregressive processes in $\mathbb C^n$ to a more general Hilbert space setting. It may be desirable to extend our results further to a Banach space setting. In a Banach space setting we no longer have the luxury of using inner products to formulate a suitable notion of cointegration. It is instead natural to define cointegration in terms of the order of integration of continuous linear functionals of an integrated process, and the cointegrating space as a subspace of the topological dual. An unpublished manuscript of the second author S18 provides some results in this direction, and also investigates relaxing the compactness condition we have imposed here on autoregressive operators. It is interesting to drop compactness because in this case the attractor space may be infinite dimensional; the difficulty is that the analytic Fredholm theorem becomes unavailable.

Beyond representation theory, research on the development of statistical procedures for analyzing cointegrated functional time series is a priority. A new manuscript by NSS19 develops a procedure for estimating the dimension of an attractor space based on sequential variance ratio tests, with apparently good small sample behavior. If the size of the literature on cointegration in the finite dimensional setting provides any indication, there remains enormous scope for further research on estimation, testing and forecasting with cointegrated functional time series.