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arxiv: 2604.07040 · v1 · submitted 2026-04-08 · 💰 econ.EM

Seasonality in Mixed Causal-Noncausal Processes

Tom\'as del Barrio Castro , Alain Hecq , Sean Telg This is my paper

Pith reviewed 2026-05-10 17:00 UTC · model grok-4.3

classification 💰 econ.EM
keywords mixed causal-noncausal autoregressive modelsseasonalitypartial fraction decompositionmoving average representationMAR model selectionseasonal rootseconometric time series
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The pith

Seasonal roots can always be isolated in the moving average representation of mixed causal-noncausal AR models.

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

The paper shows that seasonal roots remain separable in the moving average representation of mixed causal-noncausal autoregressive models. It first establishes the isolation property for purely causal and purely noncausal AR models using partial fraction decompositions, then proves the same holds when the two polynomials are multiplied together. A sympathetic reader would care because these models capture both backward-looking and forward-looking dynamics common in economics, and the result means seasonality does not introduce unexpected joint effects that would complicate specification. Monte Carlo simulations examine the consequences for model selection, and an application to COVID-19 and soybean price data illustrates the practical outcome.

Core claim

Using partial fraction decompositions, seasonal roots can always be isolated in the moving average representation of purely causal and noncausal AR models. This result extends to the MAR model, which means that no new joint seasonal effects can be generated despite the multiplicative structure of the causal and noncausal polynomials.

What carries the argument

Partial fraction decomposition of the autoregressive polynomial, which isolates seasonal roots in the moving average representation even after the causal and noncausal factors are multiplied.

If this is right

  • Standard MAR model selection procedures do not require additional adjustments for new joint seasonal interactions.
  • Seasonal patterns observed in fitted models can be attributed to either the causal part, the noncausal part, or both without interaction terms.
  • The isolation property preserves the interpretability of seasonal components when both causal and noncausal dynamics are present.
  • Monte Carlo evidence indicates that conventional selection criteria remain reliable under seasonal mixed processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of seasonal roots may allow more accurate decomposition of economic series that combine seasonality with forward-looking behavior.
  • The technique could be tested on higher-order or vector MAR models to check whether the isolation property scales.
  • Applied researchers might combine this result with existing seasonal adjustment tools without fearing hidden multiplicative seasonality.

Load-bearing premise

The partial fraction decomposition of the mixed polynomial continues to isolate seasonal roots without requiring extra restrictions that would break the moving-average representation.

What would settle it

An explicit MAR model whose derived moving-average representation contains a seasonal root absent from both the causal polynomial and the noncausal polynomial would disprove the isolation claim.

Figures

Figures reproduced from arXiv: 2604.07040 by Alain Hecq, Sean Telg, Tom\'as del Barrio Castro.

Figure 1
Figure 1. Figure 1: Simulated AR processes including ACF, PACF and smoothed periodogram Note: Frequencies have been mapped from [0, π] to 0, 1 2 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulated causal (left) and noncausal (right) AR processes with seasonal roots trajectories that are almost symmetric. However, the noncausal case reveals that bubbles can be generated which resemble the ones that are due to roots at the zero frequency. Thus, seasonal behavior is not always explicit from the trajectory. Depending on the choice of error distribution and parameter values, causal and noncausa… view at source ↗
Figure 3
Figure 3. Figure 3: Simulated MAR processes including ACF, PACF and smoothed periodogram Note: Frequencies have been mapped from [0, π] to 0, 1 2 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of the roots for different sample sizes T = 200, 500, 1000 23 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation in COVID-19 deaths for Belgium and Italy There does not appear to be a root at the zero and Nyquist frequency, but the peak in the middle (similar to Belgium) could point at the presence of complex roots. We start by estimating purely causal autoregressive models up to order pmax = 14 to identify the lag order which ensures that the residuals are free of serial correlation. Using the Bayesian Inf… view at source ↗
Figure 6
Figure 6. Figure 6: The fit of two competing MAR(2,2) models for Belgium (red dashed) the previously identified model. This result emphasizes once again the danger of identifying local instead of global maxima, which can be circumvented by performing a grid search over starting values (Bec et al., 2020) or applying simulated annealing (Cubadda et al., 2024). 5.2 Soybean Price We now focus on a financial series studied in Frie… view at source ↗
Figure 7
Figure 7. Figure 7: Soybean prices with periodogram (top row), residuals AR(5) with ACF (bottom row) peaks at most instances where the original series also peaked. This highlights the inability of a causal model to capture explosive, bubble-type behavior. The ACF confirms the absence of serial correlation. The polynomial in (25) factorizes as (1−0.852L)(1 + 0.547L)(1 + 0.125L)(1−0.539e i0.768)(1− 0.539e −i0.768), where the fi… view at source ↗
Figure 8
Figure 8. Figure 8: Simulated MAR processes and their corresponding periodograms 42 [PITH_FULL_IMAGE:figures/full_fig_p042_8.png] view at source ↗
read the original abstract

This paper investigates the role of complex and negative roots in mixed causal-noncausal autoregressive (MAR) models. Using partial fraction decompositions, we show that seasonal roots can always be isolated in the moving average representation of purely causal and noncausal AR models. We find that this result extends to the MAR model, which means that no new joint seasonal effects can be generated despite the multiplicative structure of the causal and noncausal polynomials. This results has important consequences for the MAR model selection procedure and these are extensively studied in a Monte Carlo simulation study. An empirical application on COVID-19 and soybean data illustrates the main findings of the paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper claims that seasonal roots can be isolated in the moving average representation of purely causal and noncausal AR models using partial fraction decompositions, and that this property extends to mixed causal-noncausal (MAR) models without generating new joint seasonal effects from the multiplicative structure. It studies the implications for model selection via Monte Carlo simulations and illustrates with an empirical application to COVID-19 and soybean data.

Significance. If the central algebraic result holds, it offers a clear characterization of seasonal behavior in MAR models, which could aid in model selection and interpretation in time series analysis with mixed causality. The Monte Carlo study and empirical example provide supporting evidence and practical insights. The use of partial fractions is a standard tool, and the extension to the mixed case appears straightforward as noted in the skeptic's analysis.

major comments (1)
  1. The extension of the isolation property to the MAR model via partial fractions of 1/[φ_c(L) φ_nc(F)] is load-bearing for the central claim; the manuscript should explicitly verify that no cross-frequency terms arise and state the conditions (e.g., distinct roots, validity of the two-sided MA representation) under which this holds without additional restrictions that could invalidate finite-sample behavior.
minor comments (3)
  1. Abstract: 'This results has' should be corrected to 'This result has'.
  2. Monte Carlo section: Provide more details on simulation design, including sample sizes, specific parameter values for seasonal roots, and how the isolation property was tested.
  3. Empirical application: Clarify data processing steps for the COVID-19 and soybean series and specify the seasonal frequencies examined.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the paper and the constructive suggestion regarding the central algebraic result. We address the major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: The extension of the isolation property to the MAR model via partial fractions of 1/[φ_c(L) φ_nc(F)] is load-bearing for the central claim; the manuscript should explicitly verify that no cross-frequency terms arise and state the conditions (e.g., distinct roots, validity of the two-sided MA representation) under which this holds without additional restrictions that could invalidate finite-sample behavior.

    Authors: We agree that making the verification explicit will improve clarity. In the revised manuscript we will add a short derivation immediately after the partial-fraction discussion showing that 1/[φ_c(L) φ_nc(F)] decomposes into separate sums of causal and noncausal terms with no cross-frequency components. Because φ_c(L) generates only non-negative powers and φ_nc(F) only non-positive powers in the Laurent series, and because the roots are distinct, the residues remain confined to their respective sides and no mixed-frequency terms appear. The required conditions are precisely those already stated for the MAR model to be well-defined: distinct roots (to ensure a unique partial-fraction expansion) and validity of the two-sided MA representation (which follows from the standard stationarity assumption that causal roots lie inside the unit circle and noncausal roots lie outside it). These conditions introduce no additional restrictions beyond the model definition itself. The Monte Carlo experiments already explore finite-sample behavior under exactly these conditions and confirm that the isolation property improves model selection without being undermined by sampling variability. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the isolation of seasonal roots in the MA representation of MAR models directly from the algebraic properties of partial fraction decomposition applied to the transfer function 1/[φ_c(L) φ_nc(F)]. The denominator roots are the union of those from each factor, and each simple pole contributes an independent term to the two-sided MA coefficients with no cross-frequency terms generated by multiplication. This is a standard result in rational function decomposition that holds identically for the pure causal, pure noncausal, and mixed cases without requiring parameter fitting, redefinition of the target quantity, or load-bearing self-citations. The Monte Carlo and empirical sections test consequences but do not enter the derivation. The argument is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard algebraic tools of time-series analysis; no new free parameters, ad-hoc axioms, or postulated entities are introduced in the abstract.

axioms (1)
  • standard math Partial fraction decomposition applies to the rational transfer function of an autoregressive polynomial and isolates seasonal roots in the moving-average representation.
    Invoked in the abstract as the method that separates seasonal roots for both pure and mixed cases.

pith-pipeline@v0.9.0 · 5403 in / 1315 out tokens · 66143 ms · 2026-05-10T17:00:30.115709+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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    Breidt, and R

    Andrews, B., F. Breidt, and R. Davis (2006). Maximum likelihood estimation for all-pass time series models.Journal of Multivariate Analysis 97, 1638–1659. Andrews, B. and R. Davis (2013). Model identification for infinite variance autoregressive pro- cesses.Journal of Econometrics 172(2), 222–234. Bec, F., H. Nielsen, and S. Sa¨ ıdi (2020). Mixed causal-n...

    work page 2006

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    Osborn, and R

    del Barrio Castro, T., D. Osborn, and R. Taylor (2012). On augmented hegy tests for seasonal unit roots.Econometric Theory 28(5), 1121–1143. del Barrio Castro, T., P. Rodrigues, and R. Taylor (2019). Temporal aggregation of seasonally near-integrated processes.Journal of Time Series Analysis 40, 872–886. Embrechts, P., C. Kl¨ uppelberg, and T. Mikosch (19...

    work page 2012

  3. [3]

    , j (S−1) 2 k

    (II)ω k(L) = (1−2(α k cosω k −β k sinω k)L+ (α 2 k +β 2 k)L2) which corresponds to conjugate seasonal frequencies (ω k,2π−ω k),ω k = 2πk S with associated parametersα k andβ k, for k= 1, . . . , j (S−1) 2 k . (III)ω S/2(L) = (1+αS/2L) which associates inverse rootαS/2 with the Nyquist frequencyω S/2 =π and is only defined in caseSis even. Example 4.Suppos...

    work page 2001