Causality for VARMA processes with instantaneous effects: The global Markov property, faithfulness and instrumental variables

Ignacio Gonz\'alez-P\'erez

REVIEW 4 major objections 5 minor 32 references

Causality for VARMA processes with instantaneous effects: The global Markov property, faithfulness and instrumental variables

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read m-separation in the marginalized graph of a VARMA process with instantaneous effects implies conditional independence, and under generic Gaussian coefficients the converse holds almost surely.

desk verdict Extends causal VAR results to VARMA with instantaneous effects, but Theorem 3's Markov property rests on an unproven—and on the natural reading false—finite-window projection step. read the letter →

arxiv 2501.13037 v1 pith:7GFZJBZO submitted 2025-01-22 math.ST stat.TH

Ignacio González-Pérez This is my paper · ORCID

classification math.STstat.TH MSC 62M1062D20

keywords VARMAprocessesinstantaneouseffectsglobalMarkovpropertyfaithfulnessinstrumentalvariablestotalcausalacyclicdirectedmixedgraphstimeseriescausality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper brings causal graph methods to vector autoregressive moving-average time series in which one component can influence another at the same time point. Its first result is the global Markov property: for a VARMA(p,q) process with instantaneous effects, m-separation between finite node sets in an infinite graph over the observed variables implies conditional independence in the stationary distribution. The paper then establishes almost-sure faithfulness under Gaussian innovations and continuously sampled coefficients, so that conditional independence and graph separation coincide except on a null set. Building on these two properties, it derives an instrumental-variable regression whose solution is the total causal effect and whose empirical version is consistent. A sympathetic reader would care because this extends the standard toolbox of causal discovery and effect estimation from i.i.d. data and plain VAR models to a broader, more realistic class of dependent data.

What carries the argument

The load-bearing object is the full-time marginalized ADMG G_S, defined by taking a finite 'large-enough' time window of the full-time DAG that includes both observed variables and innovations, applying the latent projection onto the observed variables, and infinitely repeating the resulting finite ADMG. m-separation is the separation criterion used throughout: a path is blocked if it contains a non-collider in the conditioning set or a collider with no descendants in it. The proof machinery rests on three transfer steps: rewriting a VARMA process with instantaneous effects as a 2d-dimensional VAR process with instantaneous effects so the VAR global Markov property applies; using the latent-projection equivalence to move m-separation in G_S back to d-separation in the full-time DAG; and, for faithfulness, expressing conditional covariances as real-analytic functions of the coefficients and applying the fact that non-zero real-analytic functions vanish on a Lebesgue-null set.

What would settle it

For a concrete VARMA(1,1) process with instantaneous effects, compute the exact stationary partial correlation between every pair of observed variables that is m-separated in G_S by some conditioning set; any nonzero partial correlation for generic coefficients would falsify Theorem 3. Conversely, if for a positive-measure set of coefficients a partial correlation vanishes despite an m-connecting path, Theorem 5 would be false.

Watch

Extended reading notes

Core claim

The central discovery is that a VARMA(p,q) process with instantaneous effects can be represented causally by an infinite acyclic directed mixed graph, the full-time marginalized ADMG G_S, whose nodes are only the observed process variables. Theorem 3 states that for any finite pairwise disjoint node sets A, B, C, m-separation of A and C by B in G_S implies conditional independence in the stationary distribution. The proof rewrites the VARMA process as a 2d-dimensional VAR process with instantaneous effects, applies the global Markov property for that broader VAR class, and uses the latent-projection separation equivalence to pass from the marginalized graph back to the full-time graph. Theorems 4 and 5 add an almost-sure faithfulness statement: when innovations are Gaussian and the coefficients and variances are drawn from an absolutely continuous distribution, every conditional independence is witnessed by separation, so the graph and distribution have the same separation and independence structure. Theorem 6 then provides an instrumental-variable moment equation whose unique solution, under a rank condition, is the total causal effect, and whose empirical minimizer is consistent.

Load-bearing premise

The construction of the infinite marginalized graph G_S assumes that the latent projection of one finite 'large-enough' time window, repeated infinitely, preserves every m-separation statement between arbitrary finite sets of the infinite graph; if that finite-to-infinite step failed for some separation statement, Theorem 3 would not hold.

Editorial extensions

If this is right

Causal discovery algorithms for acyclic directed mixed graphs become applicable to the observed variables of VARMA processes with instantaneous effects, because m-separation statements in the marginalized graph are guaranteed to be reflected in the stationary distribution.
When coefficients and variances are drawn from a distribution that is absolutely continuous with respect to Lebesgue measure, the stationary distribution is almost surely faithful, so conditional independencies estimated from data can be interpreted as m-separations.
Total causal effects along endogenous directed paths are identifiable from the instrumental-variable moment equation whenever the instrument set satisfies the stated separation and rank conditions, and the empirical estimator is consistent.
The results cover a family of processes broader than those covered by the contemporaneous global Markov property result discussed in the paper, because the MA part allows innovations other than the contemporaneous innovation of a variable into its structural equation.
Rewriting a VARMA process as a 2d-dimensional VAR with instantaneous effects means the Markov and faithfulness theorems inherit the analytic machinery of the VAR case without requiring absolute continuity of the innovation distribution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The finite-window construction of G_S suggests a practical consistency check: for small VARMA models one could enumerate all triples of node sets and compare m-separation in G_S with exact partial correlations from the stationary distribution, and a mismatch would pinpoint exactly where the infinite repetition step fails.
The almost-sure faithfulness result implies that exact coefficient cancellations are measure-zero, but near-cancellations can still create near-unfaithfulness, so structure-learning algorithms may need to treat near-zero partial correlations with care rather than relying on exact faithfulness.
The IV moment equation and rank condition are stated for the linear Gaussian setting; a natural extension would be to replace conditional covariance with rank-based or kernel conditional covariance to handle non-Gaussian innovations and nonlinear effects.
The same latent-projection-plus-periodic-repetition recipe could be applied to other stationary linear processes with finite-order representations, such as state-space models or VARMA models with exogenous inputs, to obtain Markov properties over observed variables only.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper does something genuinely new: it pushes the global Markov property, faithfulness, and instrumental-variable identification from VAR processes to VARMA processes with instantaneous effects. Theorems 2, 4, 5, and 6 are nontrivial extensions of [24], and the proofs are spelled out in detail. If the results stand, they give the causal time-series community a broader model class to work with. The author is also honest about the limits, e.g., no CLT for the IV estimator.

The soft spot is Theorem 3. The paper defines the full-time marginalized ADMG G_S by taking a finite window of length max(p,q)+2, projecting out the innovations, and then "infinitely repeating the structure." It never proves that this repetition equals the latent projection of the infinite DAG. The proof of Theorem 3 explicitly invokes Proposition 5, which requires G_S to be the latent projection of G_full. That premise is not established.

A stress-test note gives a concrete reading on which the construction is false. Its example uses A1=0, which by itself gives no common ancestors, so that specific example is wrong. But the concern is valid: take a VAR(1) with A1≠0; the finite window repeats a tile that has no edge from its last node to the first node of the next tile, so the construction omits genuine bi-directed edges across tile boundaries, e.g., S_t ↔ S_{t+1}. Thus the central reduction in Theorem 3 is unproven and, on the stated construction, actually false. The fix is likely straightforward—define G_S directly as the infinite latent projection, or prove the window construction preserves all separations—but as written the theorem lacks a rigorous basis.

The faithfulness theorems and the IV theorem inherit this, since they rely on Theorem 3. Nothing else smells fatal; the analytic-function argument in Theorem 4 is standard and looks okay.

Who should read it: researchers working on causal inference for time series, especially those building on [24]. It deserves a serious referee because the problem is important and the intended results are plausible. But it should be returned for major revision, and the referee should push hard on Definition 15 and the proof of Theorem 3.

Referee Report

4 major / 5 minor

Summary. The manuscript extends causal graphical modeling from VAR processes to VARMA processes with instantaneous effects. It defines a full-time marginalized ADMG G_S for such processes (Definition 15) and claims a global Markov property: m-separation in G_S implies conditional independence in the stationary distribution (Theorem 3). It further claims almost-sure faithfulness under generic Gaussian coefficients for VAR and VARMA processes (Theorems 4 and 5), and develops an instrumental variable regression framework for identifying and consistently estimating total causal effects (Theorem 6). The paper is explicitly positioned as an extension of Thams et al. [24] and discusses its relation to the contemporaneous work [7].

Significance. If the results were correct, the paper would fill a genuine gap by moving causal graphical modeling for time series from VAR to VARMA with instantaneous effects, and it would provide an identification and estimation framework for total causal effects. The paper is ambitious and addresses a relevant problem. However, the central graph construction in Definition 15 is not the latent projection of the infinite full-time DAG, and the counterexample below shows that Theorem 3 as stated is false. The significance is therefore conditional on a substantial correction of the construction and of the proofs that rely on it.

major comments (4)

[Definition 15, Theorem 3] Definition 15 constructs G_S by taking a finite window of length max(p,q)+2, applying the latent projection to that finite window, and then "infinitely repeating the structure." This does not equal the latent projection of the infinite full-time DAG. For the univariate VARMA(1,1) process S_t = ε_t + B ε_{t−1} with A_1 = 0 and B ≠ 0, the true infinite latent projection over {S_t} contains a bi-directed edge S_t ↔ S_{t+1} for every t, because ε_t is a common ancestor of both S_t and S_{t+1}. With max(p,q)+2 = 3, the finite window {t*−2, t*−1, t*} yields bi-directed edges only inside the tile; periodic repetition produces no edge across the tile boundary, e.g. between S_{t*+1} and S_{t*+2} in the next tile. Those adjacent nodes are then m-separated by the empty set in the constructed G_S, whereas in the true graph they are m-connected. Theorem 3 would therefore imply S_t ⊥⊥ S_{t+1}, but Cov(S_t, S_{t+1}) = B Var(ε_t) ≠ 0. The definition of G_S and the proof of Theorem 3 need to be repaired, for example by defining G_S directly as the latent projection of the infinite full-time DAG and proving the corresponding separation-equivalence lemma.
[Proof of Theorem 3, Section 3.2] The proof invokes Proposition 5 to transfer m-separation in G_S to d-separation in the full-time DAG. Proposition 5 is only applicable when G_S is the latent projection of G_full. Under Definition 15, G_S is a periodic repetition of a finite-window projection, not the latent projection of G_full, so the required premise is missing. In addition, the proof states that A and C are "d-separated" in the marginalized ADMG, where the correct criterion is m-separation; this is not merely a typo, because Proposition 5 connects m-separation in the projected ADMG with d-separation in the original DAG.
[Theorem 4, Appendix A] The faithfulness proof for VAR processes asserts that [23, Thm. 3.5] gives the equivalence X ⊥_G Y | B if and only if Cov(X,Y|B) = 0. In linear Gaussian systems this equivalence is false in general: coefficient cancellations can make the partial covariance zero even when there is a d-connecting path, as the paper's own Example 3 demonstrates. The proof needs to show that the analytic function f defined via the conditional MA(∞) representation in Eq. (18) is not identically zero on the parameter domain. As written, the statement that the map "is not trivially zero" is unsupported and is in fact the main content of the faithfulness claim.
[Theorem 5, Appendix A] Theorem 5 repeats the identically-zero issue of Theorem 4, and additionally reduces the VARMA process to a VAR(l) process Q_t whose innovations have degenerate components (P(η_i^t = 0) = 1). The proof asserts that joint normality of the stationary distribution and joint independence of the innovations suffice to apply [23, Thm. 3.5], but the hypotheses of that theorem are not verified for this degenerate VAR. Without a precise argument, the reduction does not establish that non-zero partial covariances are not identically zero as functions of the original coefficients.

minor comments (5)

[Definition 15] The role of the arbitrary time t* and the exact window length max(p,q)+2 is unexplained; if the construction is intended to be shift-invariant, this should be stated explicitly and justified.
[Remark 6] Remark 6 refers to [24, Def. 4] for a more general marginalization definition but does not reproduce that definition; the reader cannot check whether the claimed generalization is valid.
[Theorem 3 proof] In the proof of Theorem 3, the sentence before invoking Proposition 5 says A and C are "d-separated" in G_S; this should be "m-separated," since G_S is an ADMG.
[Remark 5] In the displayed equation of Remark 5, the term (I_d − A_0)^{-1} A_p appears without S_{t−p}; the intended equation presumably reads S_t = (I_d − A_0)^{-1} A_1 S_{t−1} + ... + (I_d − A_0)^{-1} A_p S_{t−p} + δ_t.
[General] There are numerous typographical and formatting issues, including inconsistent spacing in "V ARMA," the use of "ídem" instead of "ibidem" or "similarly," and occasional missing articles; these should be cleaned up in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Markov, faithfulness and IV results are derived from structural-equation assumptions and external theorems, not from their own conclusions.

full rationale

I walked the derivation chain. Theorem 2 is proved in Appendix A by adapting the proof of [24, Thm. 1]; the adaptation is not a citation-only argument. Theorem 3 rewrites the VARMA process as a 2d-dimensional VAR process and uses Proposition 5 (a standard latent-projection separation equivalence from [18]) to transfer m-separation in G_S to d-separation in the full-time DAG; the graph G_S is defined by finite-window latent projection plus periodic repetition. The finite-to-infinite repetition is asserted without proof and may be a genuine correctness gap: Proposition 5 applies to a latent projection of a finite DAG, and no lemma shows that the periodically repeated finite projection preserves m-separation for arbitrary finite A, B, C. This is a missing-premise problem, not circularity: the conclusion is not identical to the construction of G_S, and no parameter is fitted to the quantities being predicted. Theorem 4 and Theorem 5 are generic faithfulness arguments using real-analyticity and Lebesgue-null sets following [23], not fits renamed as predictions. Theorem 6 adapts [24, Thm. 5]; its proof uses Theorem 3, but again in a deductive, non-circular way. The citations to [24] and [7] are to external work by other authors; the only overlapping name is Jonas Peters in the acknowledgments as thesis supervisor, and no load-bearing claim rests on an unpublished self-citation. I therefore assign 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the structural VARMA assumption, acyclicity, Gaussianity for faithfulness, and standard analyticity and separation results. No new physical entities or fitted constants are introduced.

assumptions (6)

domain assumption The observed process S follows a VARMA(p,q) model with instantaneous effects as in Definition 14, with i.i.d. innovations having finite second moments and independent components.
All theorems assume this structural equation model for the time series.
domain assumption The full-time graph is acyclic; a sufficient condition is that the graph of instantaneous edges is acyclic (Remark 5).
The global Markov property and faithfulness results require the full-time graph to be a DAG.
domain assumption For faithfulness, innovations are Gaussian with diagonal covariance, and coefficients and variances are sampled from an absolutely continuous distribution on the open domain D∘×(0,∞)^d.
Theorems 4 and 5 rely on this distributional assumption to conclude almost-sure faithfulness.
standard math The zero set of a non-zero real analytic function on an open connected set has Lebesgue measure zero (Mityagin 2015).
Used in the faithfulness proofs to show that faithfulness violations occur on a null set.
standard math For linear Gaussian systems, d-separation is equivalent to zero partial covariance (Spirtes et al. 2001, Thm 3.5).
Used to connect graph separation with conditional independence in the Gaussian faithfulness proofs.
standard math Latent projections preserve separation statements between a DAG and its marginalized ADMG (Richardson et al. 2023, Prop 5).
Used in Theorem 3 and Theorem 5 to translate m-separation in the marginalized graph to d-separation in the full-time DAG.

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Cite this review

Pith. "Pith review of Causality for VARMA processes with instantaneous effects: The global Markov property, faithfulness and instrumental variables." pith.science (2026). https://pith.science/paper/7GFZJBZO

@misc{pith2026250113037,
  author       = {Pith},
  title        = {Pith review of: Causality for VARMA processes with instantaneous effects: The global Markov property, faithfulness and instrumental variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GFZJBZO}},
  note         = {Machine review of arXiv:2501.13037}
}

read the original abstract

Causal reasoning has gained great attention over the last half century as it allows (or at least intends) to answer questions which go above those within the capabilities of classical inferential statistics using just observational data. So far, causal research has been focused mostly on the i.i.d. setting. However, many are the situations where there exists a non-trivial dependence structure between sequential observations. Motivated by this fact, the main purpose of this work is to study causal properties of time series under the structural assumption of a VARMA model with instantaneous effects. First, the global Markov property is studied, building on existing work for VAR processes without instantaneous effects. Infinite graphs which represent the dependencies of the process are defined so that separation statements translate to conditional independencies in the stationary distribution of the process. Second, faithfulness is examined as a counterpart of this Markov property. Conditions are given so that the stationary distribution of the process is almost surely faithful to said infinite graphs. In addition, an instrumental variable regression framework is developed for VARMA models with instantaneous effects. This allows to identify and consistently estimate total causal effects.

Figures

Figures reproduced from arXiv: 2501.13037 by the authors.

**Figure 1.** Full-time DAG GI of the time series with instantaneous effects defined in Equation (2). The coefficients of the edges correspond to the coefficients of the process. Xt Yt Xt−1 Yt−1 Xt−2 Yt−2 Xt+1 Yt+1 1 2 1 2 1 6 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** Full-time DAG GII of the time series without instantaneous effects defined in Equation (3). The coefficients of the edges correspond to the coefficients of the process. As we cannot directly translate d-separation statement from GI to GII , our approach is to study how the instantaneous effects modify the correlation structure of the innovations when the process is expressed without instantaneous effects. To that e… view at source ↗

**Figure 3.** Full-time DAG GI of the time series defined in Equation (4). The coefficients of the edges correspond to the coefficients of the process. 3.3 A comment on [7] Contemporaneous to the writing of this thesis, the work by [7] was published. In this article, the authors prove the global Markov property for a certain family of time series with respect to a full-time DAG, under a series of assumptions. Their work encompass… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Full-time marginalized ADMG GI,S of the time series defined in Equation (4). consequence of [7, Thm. 1], even if the absolute-continuity assumption of the innovation distribution were added. Thus, the families of time series for which [7] and this thesis study the glob…

**Figure 5.** Figure 5: Full-time DAG GII of the time series defined in Equation (5). The coefficients of the edges correspond to the coefficients of the process. Notice how the violation of faithfulness mentioned in Example 3 comes from imposing a restriction on the coefficients of the proce…

**Figure 6.** Figure 6: Full-time marginalized ADMG GII,S of the time series defined in Equation (5). X Y Z γ α β [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Associated DAG of the process defined in Equation (6). The coefficients of the edges correspond to the coefficients of the process. is triangular inferior with a zero diagonal. Define: D := {(vech(A0), vec(A1| . . . |Ap))2 ∈ R pd2+d(d−1)/2 : det((Id − A0)λ p − A1λ p−1 …

**Figure 8.** Figure 8: Full-time DAG Gfull of the time series defined in Equation (8). The coefficients of the edges correspond to the coefficients of the process. 6 Conclusions and future work In this work we have focused on the study of causal aspects and properties for time series under t…

**Figure 9.** Figure 9: Full-time marginalized ADMG GS of the time series defined in Equation (8). identifiability condition and the expression of this estimator can be found in Theorem 6. Overall, the most notable contribution of this work is the fact that we have added instantaneous effects…

Discussion (0). Continue with ORCID to comment.

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[25]

If γ has a descendant in B it does not block p given B,
[26]

Furthermore, when taking this descendant path back to γ and then following p, γ becomes a non-collider not in B

if γ has no descendants in B, then it is an ancestor of A or C, and all the nodes in such a descendant path are non-colliders not in B. Furthermore, when taking this descendant path back to γ and then following p, γ becomes a non-collider not in B. Now we will modify p to obta...
[27]

, d } and t∈ Z draw an edge from every node in the node set P AGI (Si t)[t−p,t −1] to every node in DEGI (Si t)[t] (instantaneous de- scendants) if they did not already exist in GI

For each i∈ {1, . . . , d } and t∈ Z draw an edge from every node in the node set P AGI (Si t)[t−p,t −1] to every node in DEGI (Si t)[t] (instantaneous de- scendants) if they did not already exist in GI . This means that we draw into every node edges coming from the non-contem...
[28]

Remove all instantaneous edges from GI . Proof. Denote B = (Id− A0)−1. As Id− A0 is lower triangular with ones on the diagonal, so is B. Then the coeﬃcients of the edges with lag k∈{ 1, . . . , p } are given by BAk. For the sake of notation, denote A = Ak. Then the coeﬃcient o...
[29]

forward in time

we know that B is lower triangular with a diagonal of ones. Therefore, the equation (Id+1− A0)B = Id+1 can be written out as:         1 0 . . . . . . 0 −α 2, 1 1 . . . . . . . . . . . . . . . 0 . . . −α d, 1 . . . −α d,d −1 1 0 −α d+1, 1 . . . −α d+1,d −1−α d+1,d 1  ...
[30]

If V +⁄= A∪ B∪ C, then we expand A and C to disjoint sets A+ and C+ respectively, as per Proposition 2, such that V + = A+∪ B∪ C+ and still A+⊥G∗ C+|B
[31]

Note how A+ = A∗ [s0,t 0] = A∗∩ V 0 (idem for C+) as V +∩ V 0 =∅

Consider A∗ := A+∪ (P AG∗ (A+)∩ V 0) and C∗ := C+∪ (P AG∗ (C+)∩ V 0). Note how A+ = A∗ [s0,t 0] = A∗∩ V 0 (idem for C+) as V +∩ V 0 =∅. Also, A∗∩ C∗ =∅, as otherwise, given than A+∩ C+ =∅, there would be a common parent of A+ and C+ not in B, violating d-separation. 30 I. Gonz...
[32]

3.5] Causality for V ARMA processes with instantaneous eﬀects 35 tells us that X⊥Gf ull Y|B is equivalent to Cov(X, Y|B) = 0

Owing to the joint normality and stationarity of the proces s, [23, Thm. 3.5] Causality for V ARMA processes with instantaneous eﬀects 35 tells us that X⊥Gf ull Y|B is equivalent to Cov(X, Y|B) = 0 . Hence, if X⁄⊥Gf ull Y|B, then Cov(X, Y|B) = ∑ t∈Z ∑ d i,j,k =1 α i tβ j t Ci,...

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