Cointegration and Adjustment in the CVAR(∞) Representation of Some Partially Observed CVAR(1) Models

A multivariate CVAR(1) model for some observed variables and some unobserved variables is analysed using its infinite order CVAR representation of the observations. Cointegration and adjustment coefficients in the infinite order CVAR are found as functions of the parameters in the CVAR(1) model. Conditions for weak exogeneity for the cointegrating vectors in the approximating finite order CVAR are derived. The results are illustrated by two simple examples of relevance for modelling causal graphs.


Introduction
In a conceptual exploration of long-run causal order, Hoover (2018) applies the CVAR(1) model for the processes X t = (x 1t , . . ., x pt ) and T t = (T 1t , . . ., T mt ) , to model a causal graph.The process (X t ; T t ) is a solution to the equations where the error terms ε t are independent identically distributed (i.i.d.) Gaussian variables with mean 0 and variance Ω ε = diag(ω 11 , . . ., ω pp ) > 0, and are independent of the errors η t , which are (i.i.d.) Gaussian with mean 0 and variance Ω η .
Thus, the stochastic trends, T t are nonstationary random walks and conditions will be given below for X t to be I(1), that is, nonstationary, but ∆X t stationary.This will imply that MX t + CT t is stationary, so that X t and T t cointegrate.
The entry M ij = 0 means that x j causes x i , which is written x j → x i , and C ij = 0 means that T j → x i , and it is further assumed that M ii = 0. Note that the model assumes that there are no causal links from X t to T t , so that T t is strongly exogenous.
A simple example for three variables, x 1 , x 2 , x 3 , and a trend T, is the graph where the matrices are given by where * indicates a nonzero coefficient.Provided that I p + M has all eigenvalues in the open unit disk, it is seen that MX t+1 + CT t+1 = (I p + M)(MX t + CT t ) + Mε t+1 + Cη t+1 , determines a stationary process defined for all t.We define a nonstationary solution to (1) for t = 0, 1, . . .by Note that the starting values are It is seen that ∆X t+1 , ∆T t+1 and MX t + CT t are stationary processes for all t, and that (X t ; T t ) is a solution to Equation (1).In the following, we assume that (X t ; T t ) is defined by (2) for t = 0, 1, . . .The paper by Hoover gives a detailed and general discussion of the problems of recovering causal structures from nonstationary observations X t , or subsets of X t , when T t is unobserved, that is, X t = (X 1t ; X 2t ) where the observations X 1t are p 1 -dimensional and the unobserved processes X 2t and T t are p 2 -and m-dimensional respectively, p = p 1 + p 2 .It is assumed that there are at least as many observations as trends, that is p 1 ≥ m.
Model ( 1) is therefore rewritten as (3) Note that there is now a causal link from the observed process X 1t to the unobserved process X 2t if M 21 = 0.It follows from (3) that X 1t is I(1) and cointegrated with p 1 − m cointegrating vectors β, see Theorem 1.Therefore, ∆X 1t has an infinite order autoregressive representation, see (Johansen and Juselius 2014, Lemma 2), which is written as where the operator norm The matrices α and β are p 1 × m of rank m, and Here, the prediction errors ν β t+1 are i.i.d.N p 1 (0, Σ), where Σ is calculated below.The representation of X 1t , similar to (2), is where Here, β ⊥ is a p 1 × (p 1 − m) matrix of full rank for which β β ⊥ = 0, and similarly for α ⊥ .This shows that X 1t is a cointegrated I(1) process, that is, X 1t is nonstationary, while β X 1t and ∆X 1t are stationary.
A statistical analysis, including estimation of α, β, and Γ, can be conducted for the observations X 1t , t = 1, . . .T, using an approximating finite order CVAR, see Saikkonen (1992) and Saikkonen and Lütkepohl (1996).Hoover (2018) investigates, in particular, whether weak exogeneity for β in the approximating finite order CVAR, that is, a zero row in α, is a useful tool for finding the causal structure in the graph.
The present note solves the problem of finding expressions for the parameters α and β in the CVAR(∞) model (4) for the observation X 1t , as functions of the parameters in model (3), and finds conditions on these for the presence of a zero row in α, and hence weak exogeneity for β in the approximating finite order CVAR.

The Assumptions and Main Results
First, some definitions and assumptions are given, then the main results on α and β are presented and proved in Theorems 1 and 2. These results rely on Theorem A1 on the solution of an algebraic Riccati equation, which is given and proved in the Appendix A.
In the following, a k Throughout, E t (.) and Var t (.) denote conditional expectation and variance given the sigma-field F 0,t = σ{X 1,s , 0 ≤ s ≤ t}, generated by the observations.Assumption 1.In Equation (3), it is assumed that (i) ε 1t , ε 2t , and η t are mutually independent and i.i.d.Gaussian with mean zero and variances Ω 1 , Ω 2 , and Ω η , where Ω 1 and Ω 2 are diagonal matrices, (ii 2), such that ∆X t and MX t + CT t are stationary.
Assumption 1(ii) on M 11 , M 22 and M is taken from Hoover (2018) to ensure that, for instance, the process X t given by the equations X t = (I p + M)X t−1 + input, is stationary if the input is stationary, such that the nonstationarity of X t in model ( 3) is created by the trends T t , and not by the own dynamics of X t as given by M. It follows from this assumption that M is nonsingular, because I p + M is stable, and similarly for M 11 and

The Main Results
The first result on β is a simple consequence of model (3).
Theorem 1. Assumption 1 implies that the cointegrating rank is r = p 1 − m, and that the coefficients β and β ⊥ in the CVAR(∞) representation for X 1t , see (4), are given for p 1 > m as For p 1 = m, β ⊥ has rank p 1 , and there is no cointegration: α = β = 0.

Proof of Theorem of 1.
From the model Equation (3), it follows, by eliminating X 2t from the first two equations, that Solving for the nonstationary terms gives Multiplying by By Assumption 1(i), C 1.2 has rank m, so that β has rank p 1 − m, which proves (6).
The result for α is more involved and is given in Theorem 2. The proof is a further analysis of ( 7) and involves first, the representation X 1t in terms of a sum of prediction errors 5), and second, a representation of E(T t |F 0,t ) = E(T t |X 10 , . . ., X 1t ) as the (weighted) sum of the prediction errors ν 0t = ∆X 1t − E(∆X 1t |F 0,t−1 ).The second representation requires a result from control theory on the solution of an algebraic Riccati equation, together with some results based on the Kalman filter for the calculation of the conditional mean and variance of the unobserved processes X 2t , T t given the observations X 0s , 0 ≤ s ≤ t.These are collected as Theorem A1 in the Appendix A.
For the discussion of these results, it is useful to reformulate (3) by defining the unobserved variables and errors and the matrices Then, (3) becomes One can then show, see Theorem A1, that based on properties of the Gaussian distribution, a recursion can be found for the calculation of ), using the matrices in ( 8) and ( 9), by the equations Some It then follows from results from control theory, that V = lim t→∞ Var t (T * t ) exists and satisfies the algebraic Riccati equation Moreover, the prediction errors

and the prediction errors
where Comparing the representation (5) for X 1t and ( 14) for E t (T t ) gives a more precise relation between the coefficients of the nonstationary terms in (7).The main result of the paper is to show how this leads to expressions for the coefficients α and α ⊥ as functions of the parameters in model (3).
Theorem 2. Assumption 1 implies, that the coefficients α and α ⊥ in the CVAR(∞) representation of X 1t are given for p 1 > m as where Proof of Theorem 2. The left hand side of ( 7) has two nonstationary terms.The observation X 1t is represented in (5) in terms of a random walk in the prediction errors ν β i , plus a stationary term, and T t is a random walk in η i .Calculating the conditional expectation given the sigma-field F 0,t , T t is replaced by E t (T t ), which in ( 14) is represented as a weighted sum of ν 0i .Thus, the conditional expectation of (7) gives where the right hand side is bounded in mean: Setting t = [nu] and dividing by n 1/2 , it follows from (5) that where W ν (u) is the Brownian motion generated by the i.i.d.prediction errors ν β t .From ( 14), it can be proved that This follows by replacing V t , Σ t by V, Σ, because for Next we can replace ν 0t by ν β t as follows: For t = 0, 1, . . . the sum is measurable with respect to both F β t and F 0t , such that Then and therefore Finally, setting t = [nu] and normalizing (17) by n −1/2 , it follows that in the limit This relation shows that the coefficient to W ν (u) is zero, so that α ⊥ can be chosen as and therefore α = Σ(M 12 V 2T + C 1 V TT ) ⊥ which proves (15).

Two Examples of Simplifying Assumptions
It follows from Theorem 2 that in order to investigate a zero row in α, the matrix V is needed.This is easy to calculate from the recursion (11), for a given value of the parameters, but the properties of V are more difficult to evaluate.In general, α does not contain a zero row, but if M 12 V 2T = 0, the expressions for α and α ⊥ simplify, so that simple conditions on M 12 and C 1 imply a zero row in α and hence give weak exogeneity in the statistical analysis of the approximating finite order CVAR.This extra condition, M 12 V 2T = 0, implies that

and
( Thus, a condition for a zero row in α is because Ω 1 = diag(ω 1 , . . ., ω p 1 ).This is simple to check by inspecting the matrices M 12 and C 1⊥ in model (3).In the next section, two cases are given, where such a simple solution is available.
Thus, α has a zero row if C 1⊥ has a zero row.
An example of M 12 = 0 is the chain T → x 1 → x 2 → x 3 , where X 1 = {x 1 , x 2 , x 3 } is observed and X 2 = 0, and hence M 12 = 0 and C 2 = 0.Then, because T → x 1 Thus, the first row of C 1⊥ is a zero row, such that x 1 is weakly exogenous.
To formulate the next case, a definition of strong orthogonality of two matrices is introduced.
Definition 1.Let A be a k × k 1 matrix and B a k × k 2 matrix.Then, A and B are called strongly orthogonal if A DB = 0 for all diagonal matrices D, or equivalently if A ji B j = 0 for all i, j, .
Thus, if A ji = 0, we assume that row j of B is zero, and if B j = 0, row j of A is zero.A simple example is Thus, the definition means that if two matrices are strongly orthogonal, it is due to the positions of the zeros and not to linear combination of nonzero numbers being zero.
Thus, in particular if M 12 and C 1 are strongly orthogonal, and if T causes a variable in X 1 , then X 2 does not cause that variable.The expression for V simplifies in the following case.
Proof of Lemma 1.We first prove that V t is blockdiagonal for t = 0. From (2), it follows that Thus, if Φ denotes the variance of (X 10 ; X 20 ) , then and hence blockdiagonal.Assume, therefore, that V t =blockdiag(V t22 ; V tTT ) and consider the expression for V t+1 , see (11).In this expression, Q * is block diagonal (because C 2 = 0) and Q * V t Q * and Ω * are block diagonal, and the same holds for is block diagonal.To simplify the notation, define the normalized matrices Then, by assumption, A direct calculation shows that and hence V t+1 are block diagonal.Taking the limit for t → ∞, it is seen that also V is block diagonal.
Consider again the chain T → x 1 → x 2 → x 3 , but assume now that x 2 is not observed.Thus, X 1 = {x 1 , x 3 } and X 2 = {x 2 }.Here, T causes x 1 , and x 2 causes x 3 , so that Note that M 12 DC 1 = 0 for all diagonal D because T and X 2 cause disjoint subsets of X 1 .This, together with C 2 = 0, implies that V is block diagonal and that (21) holds.Thus, x i is weakly exogenous, e i α = 0, if e i C 1⊥ = e i 0 * = 0.

Conclusions
This paper investigates the problem of finding adjustment and cointegrating coefficients for the infinite order CVAR representation of a partially observed simple CVAR(1) model.The main tools are some classical results for the solution of the algebraic Riccati equation, and the results are exemplified by an analysis of CVAR(1) models for causal graphs in two cases where simple conditions for weak exogeneity are derived in terms of the parameters of the CVAR(1) model.LR(1995, (4.1.3) For |λ| > 1, using Assumption 1(ii), it is seen that rank(M(λ)) = rank(I p 2 + M 22 − λI p 2 ) + rank(I m − λI m ) = p 2 + m, because λ is not an eigenvalue of the stable matrix I p 2 + M 22 , when |λ| > 1.