{"id":"6fbc909c-1392-430e-87fe-9f22927feafb","arxiv_id":"2501.13037","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For VARMA models with instantaneous effects, graph separation in a constructed infinite graph implies conditional independence, and an IV regression identifies total causal effects.","lead":"This paper proves that causal graphs for VARMA time series with instantaneous effects satisfy the global Markov property and gives conditions for faithfulness. It also develops an instrumental variable method to identify and consistently estimate total causal effects in such models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 15's finite-window latent projection is not proven to equal the infinite latent projection; periodic repetition can omit genuine edges, so Theorem 3's Markov property lacks a rigorous premise.","rationale":"The reader's weakest assumption is exactly the finite-to-infinite projection step in Definition 15. I agree: Theorem 3 depends on translating m-separation in G_S into d-separation in the full-time DAG, but G_S is not defined as the latent projection of the infinite graph, and the asserted preservation is not proved. The univariate MA(1)-type example shows that one concrete implementation of 'repeating the structure' (periodic tiling) produces a graph with spurious m-separations and would imply false conditional independencies. This is a genuine, load-bearing gap in the proof as written, though likely repairable by redefining G_S as the full infinite latent projection (or equivalently by proving the finite-window repetition lemma) and by fixing a secondary algebraic slip in the proof's VAR embedding, where C0's top-right block should be I_d to include the contemporaneous innovation ε_t in the S-block equation. These issues do not refute the theorem, but they do mean the central Markov property is not yet rigorously established. The existing CONDITIONAL verdict therefore stands unchanged.","tokens_in":31736,"tokens_out":21071,"duration_ms":226244,"concrete_test":"Compute the graph produced by Definition 15 for the univariate VARMA(1,1) process S_t = ε_t + B ε_{t−1} with B≠0: take the finite window {t*−2, t*−1, t*}, form its latent projection over the three S nodes, and repeat the resulting structure periodically. Check whether the infinite graph contains S_t ↔ S_{t+1} for every t, in particular across tile boundaries. If not, the construction has omitted a genuine latent-projection edge; compare with Cov(S_t, S_{t+1}) = B Var(ε_t) ≠ 0 to confirm the omission is not innocuous. Alternatively, if the intended repetition is non-periodic, write down the formal edge rules of G_S and prove that m-separation in G_S coincides with m-separation in the ordinary infinite latent projection of G_full for all finite A, B, C.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3 transfers m-separation in the marginalized ADMG G_S to d-separation in the full-time DAG via Proposition 5. Proposition 5 applies only when G_S is the latent projection of G_full; but Definition 15 builds G_S by projecting a finite window and then 'infinitely repeating the structure'. No lemma states that this repetition preserves m-separation for every finite A, B, C. Under the natural periodic reading the construction is false. Take the univariate VARMA(1,1) process S_t = ε_t + B ε_{t−1} (A1=0, B≠0). The true infinite latent projection over S contains a bi-directed edge S_t ↔ S_{t+1} for every t, because ε_t is a common ancestor of both. Definition 15 with max(p,q)+2 = 3 (e.g. times t*−2, t*−1, t*) projects a three-node tile with consecutive bi-directed edges inside the tile; repeating with period 3 gives no edge across the tile boundary, e.g. between S_{kL+2} and S_{(k+1)L}. These adjacent nodes are then m-separated by ∅ in the constructed G_S, while in the true graph they are m-connected. Theorem 3 would therefore imply S_{kL+2} ⊥⊥ S_{(k+1)L}, but Cov(S_t, S_{t+1}) = B Var(ε_t) ≠ 0, so the independence claim is false. A different reading of 'repeating' may avoid this, but then the definition and the required separation-equivalence lemma are missing. As written, the central reduction in the proof of Theorem 3 is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":32029,"tokens_out":7186,"duration_ms":85123,"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":[{"comment":"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.","section":"Definition 15, Theorem 3"},{"comment":"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.","section":"Proof of Theorem 3, Section 3.2"},{"comment":"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.","section":"Theorem 4, Appendix A"},{"comment":"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.","section":"Theorem 5, Appendix A"}],"minor_comments":[{"comment":"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.","section":"Definition 15"},{"comment":"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.","section":"Remark 6"},{"comment":"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.","section":"Theorem 3 proof"},{"comment":"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.","section":"Remark 5"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The central construction of G_S in Definition 15 is demonstrably not the latent projection of the infinite full-time DAG, and the counterexample in the report shows that Theorem 3 as stated is false. This is a load-bearing error, not a presentation issue. I recommend major revision rather than rejection because the paper's overall program is coherent and could likely be repaired by defining G_S as the actual infinite latent projection and proving the missing separation-equivalence lemma, then re-checking the proofs of Theorems 4 and 5, which also contain unsupported identically-zero arguments. The editor may also wish to verify the paper's description of the relationship to [7]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis 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.\n\nThe 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.\n\nA 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":32559,"tokens_out":6122,"would_cite":false,"duration_ms":57778,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62M10","62D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["VARMA processes","instantaneous effects","global Markov property","faithfulness","instrumental variables","total causal effects","acyclic directed mixed graphs","time series causality"],"falsifier":"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.","tokens_in":20,"feed_emoji":"📈","tokens_out":12031,"duration_ms":222409,"temperature":0.7,"pith_summary":"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.","feed_headline":"Graph separation equals conditional independence in VARMA time series","feed_subtitle":"A graph over observed variables carries the Markov property; generic coefficients make it faithful.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the base global Markov property for VAR processes without instantaneous effects and the instrumental-variable regression framework that this work extends.","marker":"[24]"},{"why":"Gives a contemporaneous global Markov property for a different class of time series; the paper argues its Theorem 3 is not a consequence of it because VARMA MA terms allow innovations other than the contemporaneous one into structural equations.","marker":"[7]"},{"why":"Provides the latent-projection construction and the separation equivalence used to pass from the marginalized ADMG back to the full-time DAG in Theorem 3.","marker":"[18]"},{"why":"Defines m-separation and the augmented graph for acyclic directed mixed graphs, the separation criterion that Theorem 3 uses.","marker":"[17]"},{"why":"Gives the moralization characterization of d-separation used in the induction proof of the VAR global Markov property that Theorem 2 adapts.","marker":"[12]"},{"why":"Supplies the linear-Gaussian faithfulness template and the theorem connecting zero conditional covariance with conditional independence in the stationary distribution, used in Theorems 4 and 5.","marker":"[23]"},{"why":"Provides the stationarity and invertibility conditions for VAR and VARMA processes that Definition 14 relies on.","marker":"[6]"}],"fun_headline_variants":["Graph separation equals independence in VARMA time series","VARMA causality: Markov property, faithfulness, instruments","Instantaneous VARMA effects now have causal graphs","Faithful VARMA graphs and instrumental variable estimation","Total effects from VARMA via instrumental variables"],"cache_read_input_tokens":34688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Graph separation equals independence in VARMA time series","VARMA causality: Markov property, faithfulness, instruments","Instantaneous VARMA effects now have causal graphs","Faithful VARMA graphs and instrumental variable estimation","Total effects from VARMA via instrumental variables"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1248,"prompt_tokens":955,"completion_tokens":293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":222}},"tokens_in":571,"tokens_out":293,"duration_ms":2916,"temperature":1.0,"reasoning_tokens":222,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:29:08.095506+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Identifying Causal Effects using Instrumental Time Series: Nuisance IV and Correcting for the Past","cited_arxiv_id":"2203.06056","evidence_quote":"Supplies the base global Markov property for VAR processes without instantaneous effects and the instrumental-variable regression framework that this work extends."},{"cited_title":"In: The 40th Conference on Uncertainty in Artificial Intelligence ( 2024)","cited_arxiv_id":null,"evidence_quote":"Gives a contemporaneous global Markov property for a different class of time series; the paper argues its Theorem 3 is not a consequence of it because VARMA MA terms allow innovations other than the contemporaneous one into structural equations."},{"cited_title":"The Annals of Statistics 51(1), 334–361 (2023)","cited_arxiv_id":null,"evidence_quote":"Provides the latent-projection construction and the separation equivalence used to pass from the marginalized ADMG back to the full-time DAG in Theorem 3."},{"cited_title":"Scandinavian Journal of Statistics 30(1), 145–157 (2003)","cited_arxiv_id":null,"evidence_quote":"Defines m-separation and the augmented graph for acyclic directed mixed graphs, the separation criterion that Theorem 3 uses."},{"cited_title":"MIT press (2001) Causality for V ARMA processes with instantaneous effects 23","cited_arxiv_id":null,"evidence_quote":"Supplies the linear-Gaussian faithfulness template and the theorem connecting zero conditional covariance with conditional independence in the stationary distribution, used in Theorems 4 and 5."}],"review_version":1}