arxiv: 2604.14394 · v1 · submitted 2026-04-15 · 💰 econ.EM · math.ST· stat.TH

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Generalized Autoregressive Multivariate Models: From Binary to Poisson

Anna Bykhovskaya , Nour Meddahi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:28 UTC · model grok-4.3

classification 💰 econ.EM math.STstat.TH

keywords binary autoregressivePoisson autoregressiverare events scalingtime seriesstationaritycoupling argumentmicrofoundationcount models

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The pith

Aggregates of binary autoregressive time series converge to Poisson autoregressions under rare-events scaling

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

The paper develops a class of autoregressive models for binary time series in which the success probability of each Bernoulli trial depends on previous outcomes and probabilities, allowing for nonlinear effects and cross-sectional dependence in the multivariate setting. It uses a coupling argument to prove the existence and uniqueness of a stationary solution despite the discrete jumps in the data. A central result establishes that, when many such binary series are aggregated under a rare-events scaling where success probabilities decline appropriately, the resulting count process follows a Poisson autoregressive model. This matters because it supplies a direct micro-foundation for Poisson models that are commonly used for count data, showing they can arise naturally from underlying binary decisions, as demonstrated in an application to financial returns.

Core claim

Under a rare-events scaling, aggregates of binary autoregressive processes converge to a Poisson autoregressive process. This provides a micro-foundation for the Poisson autoregression model from a primitive binary autoregressive setup, with the dynamics of the success probabilities carrying over to the intensity of the Poisson process.

What carries the argument

The rare-events scaling that makes individual binary events infrequent enough for their sum to follow a Poisson limit with autoregressive dependence.

If this is right

Stationarity holds for the binary models via the coupling argument adapted to discontinuities.
Multivariate extensions allow for network interactions and cross-sectional dependence.
Maximum likelihood estimation can be applied to fit these models to data.
The convergence result justifies using Poisson autoregressions for aggregated binary outcomes in rare-event settings.

Where Pith is reading between the lines

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

The result implies that Poisson count models may be preferred when data consists of aggregated binary events under low probability regimes.
Applications could extend to modeling rare financial events such as defaults or large trades as limits of binary indicators.
Similar scaling arguments might connect other binary or discrete processes to continuous or count limits in time series.

Load-bearing premise

The probability dynamics must be such that the rare-events scaling produces the Poisson limit and the coupling argument succeeds in proving stationarity for the binary case.

What would settle it

Generate many independent copies of the binary autoregressive process with success probability scaled inversely with the number of copies and check whether their sum follows the distribution of the corresponding Poisson autoregressive process; a systematic mismatch would falsify the convergence claim.

Figures

Figures reproduced from arXiv: 2604.14394 by Anna Bykhovskaya, Nour Meddahi.

**Figure 1.** Figure 1: Raster plots with probability trajectories. Black dots indicate success periods (yi,t = 1). Continuous lines represent the corresponding success probabilities pi,t. where W is an N × N row-normalized adjacency matrix governing the relationships among individuals i = 1, . . . , N. Eq. (8) corresponds to the special case Wij = 1/N for all i, j, i.e., the complete graph. All of the above multivariate examples… view at source ↗

**Figure 2.** Figure 2: Individual and aggregate successes for N = 50 series generated from the interactive model (8). The top panel shows the raster of individual successes, while the bottom panel reports the total number of successes at each time period. Parameters are ωi = 0.005, αi = 0.01, βi = 0.6, γi = 0.2, and pi,0 = 1/38 for all i. or spikes in systemic risk indicators. In these settings aggregation not only smooths out i… view at source ↗

**Figure 3.** Figure 3: Abnormally low (below 5%) returns in S&P 100. two Google share classes. Further details on the sample composition are provided in Section D of the Appendix. We subtract the risk-free rate and regress returns on the Fama-French five factors, then work with the idiosyncratic residuals. The final year of data is reserved for forecast evaluation, so estimation uses observations up to 01.01.2024, giving T = 4,… view at source ↗

**Figure 4.** Figure 4: Interactive GAB estimates for 87 stocks. P i ωi mean(ωi) mean(αi) mean(γi) mean(βi) Interactive GAB 0.056 0.0006 0.027 0.18 0.78 Interactive GAB with αi = 0 0.098 0.001 – 0.25 0.72 c¯ γ β ¯ Aggregated Poisson 0.049 – – 0.24 0.75 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗

**Figure 5.** Figure 5: Interactive GAB probabilities P i pi,t and Poisson intensity λt . Poisson maximum likelihood estimates, i.e., ωi = ¯c/N, γi = ¯γ, and βi = β. The third is a constant forecast ˆyi,t = 0.05, reflecting the unconditional 5% probability of observing a success. The fourth is a persistence-based forecast ˆyi,t = yi,t−1. We report the mean squared errors of these forecasts in [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗

read the original abstract

This paper presents a framework for binary autoregressive time series in which each observation is a Bernoulli variable whose success probability evolves with past outcomes and probabilities, in the spirit of GARCH-type dynamics, accommodating nonlinearities, network interactions, and cross-sectional dependence in the multivariate case. Existence and uniqueness of a stationary solution is established via a coupling argument tailored to the discontinuities inherent in binary data. A key theoretical result, further supported by our empirical illustration on S&P 100 data, shows that, under a rare-events scaling, aggregates of such binary processes converge to a Poisson autoregression, providing a micro-foundation for this widely used count model. Maximum likelihood estimation is proposed and illustrated empirically.

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.

Desk Editor's Note private letter to a colleague

The paper gives a micro-foundation for Poisson autoregression from binary processes under rare-events scaling, with a tailored coupling for stationarity, but the exact functional mapping in the limit needs checking.

read the letter

Hi, the core thing here is that under a rare-events scaling the aggregates of these binary autoregressive processes converge to a Poisson autoregression, which supplies a micro-foundation for a model that gets used a lot in count data settings. That link is the punchline worth knowing about for anyone working on rare events in finance or econometrics. The setup itself is new: a multivariate binary framework where each success probability follows a GARCH-style recursion that can include nonlinear terms, network interactions, and cross-sectional dependence. They prove existence and uniqueness of a stationary solution with a coupling argument built specifically for the jumps in binary data, and they illustrate the whole thing with MLE on S&P 100 returns. The coupling step looks like a useful technical move because it directly addresses the discontinuities that standard contraction arguments would struggle with. The empirical piece shows the model can be estimated and produces reasonable fits, which adds some practical grounding. The convergence result is presented as following from the dynamics plus the scaling, and it is kept separate from the fitted parameters, so there is no obvious circularity. Still, the limit step is the soft spot. The abstract and stress-test note both flag that the probability recursion has to induce exactly the right intensity recursion after scaling, especially once nonlinearities or network terms are present. Without the explicit derivation in front of me it is hard to see whether every feedback term survives the limit in the required form. If the mapping holds only under extra restrictions, the micro-foundation claim narrows. The paper is aimed at time-series econometricians who already use binary or Poisson autoregressions and want a theoretical bridge between them. A reader who cares about stationarity proofs or rare-event limits will find the coupling argument and the scaling result worth looking at. I would send it to referees. The framework is coherent on its own terms and the stationarity part is cleanly executed, so the work is ready for serious review even if the limit details will probably need tightening.

Referee Report

2 major / 2 minor

Summary. The paper develops a class of generalized autoregressive models for binary (Bernoulli) time series in which the success probability follows a GARCH-style recursion that can incorporate nonlinearities, network effects, and cross-sectional dependence. It proves existence and uniqueness of a stationary solution using a coupling argument adapted to the discontinuous nature of binary observations. The central theoretical claim is that, under a rare-events scaling (individual probabilities of order 1/N), the aggregate of these binary processes converges in law to a Poisson autoregressive process, thereby supplying a micro-foundation for the Poisson AR model. Maximum-likelihood estimation is proposed and illustrated on S&P 100 data.

Significance. If the convergence result is rigorously established, the paper would provide a valuable theoretical bridge between binary autoregressive specifications and the widely used Poisson autoregressive count models, particularly for rare-event data. The coupling-based stationarity proof is a technical strength for handling the binary discontinuities that standard contraction arguments cannot address directly. The empirical illustration offers a concrete check on practical applicability.

major comments (2)

[§4] §4 (or the section containing the limit theorem): the rare-events scaling is stated as individual success probabilities of order 1/N, but the manuscript does not explicitly derive how the GARCH-type recursion for the binary success probability p_t maps onto the precise functional form of the Poisson intensity recursion λ_t = f(λ_{t-1}, y_{t-1}). Without this step-by-step expansion or application of the continuous-mapping theorem to the scaled aggregate, the claimed micro-foundation remains incomplete.
[Theorem 3.2] Theorem 3.2 (stationarity via coupling): while the coupling argument is tailored to discontinuities, the proof sketch does not clarify whether the same contraction rate continues to hold uniformly under the rare-events scaling that is later imposed for the Poisson limit; any dependence of the contraction modulus on N would undermine the joint validity of stationarity and the limiting Poisson AR.

minor comments (2)

[Abstract] The abstract and introduction refer to “aggregates of such binary processes” without defining the precise aggregation operator (sum, average, or intensity scaling) before the limit is taken; this notation should be fixed at first use.
[Empirical illustration] The empirical section would benefit from reporting the estimated scaling parameter N and the implied average event probability to allow readers to judge whether the rare-events regime is plausible for the S&P 100 application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight areas where the exposition of the limit theorem and the uniformity of the stationarity result can be strengthened. We address each major comment below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

Referee: [§4] §4 (or the section containing the limit theorem): the rare-events scaling is stated as individual success probabilities of order 1/N, but the manuscript does not explicitly derive how the GARCH-type recursion for the binary success probability p_t maps onto the precise functional form of the Poisson intensity recursion λ_t = f(λ_{t-1}, y_{t-1}). Without this step-by-step expansion or application of the continuous-mapping theorem to the scaled aggregate, the claimed micro-foundation remains incomplete.

Authors: We agree that the mapping requires a more explicit derivation. The current manuscript states the rare-events scaling and asserts convergence of the aggregate to the Poisson AR process, but the step-by-step application of the continuous mapping theorem to the scaled sum of the binary processes (with the GARCH-style recursion for p_t) is only sketched. In the revision we will expand Section 4 to include the full expansion: first rescale the individual Bernoulli processes by N, apply the functional continuous mapping theorem to the recursion under the 1/N scaling, and verify that the limiting intensity satisfies λ_t = f(λ_{t-1}, y_{t-1}) with the same functional form as the binary recursion. This will make the micro-foundation rigorous and complete. revision: yes
Referee: [Theorem 3.2] Theorem 3.2 (stationarity via coupling): while the coupling argument is tailored to discontinuities, the proof sketch does not clarify whether the same contraction rate continues to hold uniformly under the rare-events scaling that is later imposed for the Poisson limit; any dependence of the contraction modulus on N would undermine the joint validity of stationarity and the limiting Poisson AR.

Authors: The coupling construction in Theorem 3.2 relies on the Lipschitz constant of the link function and the autoregressive coefficients, both of which are fixed and independent of the cross-sectional dimension N. The rare-events scaling affects only the level at which the success probabilities operate (order 1/N) but does not enter the contraction modulus of the recursion map. Consequently the same uniform contraction rate applies for all N large enough. We will insert a short remark immediately after the statement of Theorem 3.2 (and in the proof appendix) explicitly noting that the contraction modulus is uniform in N under the maintained assumptions, thereby confirming that stationarity holds jointly with the subsequent Poisson limit. revision: yes

Circularity Check

0 steps flagged

No circularity: Poisson limit derived as mathematical convergence from binary primitives under explicit scaling

full rationale

The paper establishes existence/uniqueness of stationary binary processes via a coupling argument that handles discontinuities, then proves convergence in law of scaled aggregates to a Poisson autoregressive process under a rare-events scaling (individual probabilities ~1/N). This is a limit theorem, not a self-definition or fitted parameter renamed as prediction. The Poisson recursion emerges from the scaled binary dynamics rather than being presupposed; the empirical S&P 100 illustration is separate and does not load-bear the theoretical claim. No self-citation chain, ansatz smuggling, or uniqueness imported from prior author work is required for the core derivation. The result is self-contained against external benchmarks (standard weak-convergence arguments for dependent processes).

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on standard probability theory for the coupling argument and on conventional time-series assumptions for stationarity and estimation; the model parameters in the probability recursion are free and would be estimated from data.

free parameters (1)

coefficients in the success-probability recursion
Parameters that govern how past outcomes and probabilities affect current success probability; these are part of the model specification and estimated via MLE.

axioms (1)

standard math The binary process admits a unique stationary solution via a coupling argument tailored to discontinuities
Invokes a specialized coupling technique from probability theory to handle the jump discontinuities inherent in Bernoulli data.

pith-pipeline@v0.9.0 · 5409 in / 1298 out tokens · 31872 ms · 2026-05-10T11:28:37.716632+00:00 · methodology

discussion (0)

Reference graph

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