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arxiv: 2604.12093 · v1 · submitted 2026-04-13 · 🧮 math.ST · stat.TH

QBIC of SEM for jump-diffusion processes based on high-frequency data

Shogo Kusano , Masayuki Uchida This is my paper

Pith reviewed 2026-05-10 14:56 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords structural equation modelingjump-diffusion processesmodel selectionquasi-Bayesian information criterionhigh-frequency dataconsistencyquasi-likelihood
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The pith

A quasi-Bayesian information criterion selects the correct SEM for jump-diffusion processes with high-frequency data.

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

This paper introduces a quasi-Bayesian information criterion for choosing among candidate structural equation models when observations come from a jump-diffusion process sampled at high frequency. The authors prove that the criterion achieves model-selection consistency, identifying the true model with probability approaching one as the number of observations increases. A sympathetic reader would care because SEM is used to recover relationships among latent variables in time series that contain jumps, and a consistent selector reduces the risk of choosing misspecified models in applications such as finance or physics. The approach combines quasi-likelihood methods with a Bayesian-style penalty tailored to the jump-diffusion setting.

Core claim

We propose a quasi-Bayesian information criterion (QBIC) for the SEM and show that the proposed criterion has model-selection consistency.

What carries the argument

The quasi-Bayesian information criterion (QBIC), derived from the quasi-likelihood of the SEM parameters under a jump-diffusion model, with a penalty term that enforces consistency in selection.

If this is right

  • The selected model will correctly represent the latent variable relationships driving the observed jump-diffusion process.
  • As the number of high-frequency observations grows, the probability of selecting an over- or under-specified SEM approaches zero.
  • Practitioners can apply the QBIC directly to high-frequency data without needing to estimate the full likelihood.

Where Pith is reading between the lines

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

  • The same QBIC construction could be tested on other jump processes such as Levy-driven models if suitable quasi-likelihoods exist.
  • This selection tool may improve causal analysis in econometric models that combine SEM with stochastic volatility and jumps.
  • One could examine finite-sample performance through Monte Carlo experiments that vary jump intensity and observation frequency.

Load-bearing premise

The candidate SEMs and the true jump-diffusion process satisfy the regularity and identifiability conditions required for the quasi-likelihood and the consistency theorem to hold.

What would settle it

A simulation or real-data example in which the QBIC selects an incorrect model with probability bounded away from zero, even as the high-frequency sampling rate tends to infinity, would falsify the consistency claim.

Figures

Figures reproduced from arXiv: 2604.12093 by Masayuki Uchida, Shogo Kusano.

Figure 1
Figure 1. Figure 1: The path diagram of the true model at time t. 4.2. Model 1. Set p1 = 5, p2 = 10, k1 = 1, k2 = 2 and q1 = 32. Assume that Λ θ 1,1 =  1 θ (1) 1 θ (2) 1 θ (3) 1 θ (4) 1 ⊤ , [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 4.3. Model 2. Set p1 = 5, p2 = 10, k1 = 1, k2 = 2 and q2 = 33. Supposed that Λ θ 1,2 =  1 θ (1) 2 θ (2) 2 θ (3) 2 θ (4) 2 ⊤ , Λ θ 2,2 = [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: The path diagram of Model 1 at time t. where θ (i) 2 for i = 1, . . . , 8 and 10, . . . , 15 are not zero. Furthermore, we assume that Σθ ξξ,2 = θ (16) 2 , Σθ δδ,2 = Diag θ (17) 2 , θ(18) 2 , θ(19) 2 , θ(20) 2 , θ(21) 2 ⊤ , Σθ εε,2 = Diag θ (22) 2 , θ(23) 2 , θ(24) 2 , θ(25) 2 , θ(26) 2 , θ(27) 2 , θ(28) 2 , θ(29) 2 , θ(30) 2 , θ(31) 2 ⊤ and Σθ ζζ,2 = Diag θ (32) 2 , θ(33) 2 ⊤ , where θ (i) 2 for i = 16… view at source ↗
Figure 3
Figure 3. Figure 3: The path diagram of Model 2 at time t. 4.4. Model 3. Set p1 = 5, p2 = 10, k1 = 1, k2 = 1 and q3 = 31. Assume that Λ θ 1,3 =  1 θ (1) 3 θ (2) 3 θ (3) 3 θ (4) 3 ⊤ and Λ θ 2,3 =  1 θ (5) 3 θ (6) 3 θ (7) 3 θ (8) 3 θ (9) 3 θ (10) 3 θ (11) 3 θ (12) 3 θ (13) 3 ⊤ , Γ θ 3 = θ (14) 3 , where θ (i) 3 for i = 1, . . . , 14 are not zero. Suppose that Σθ ξξ,3 = θ (15) 3 , Σθ δδ,3 = Diag θ (16) 3 , θ(17) 3 , θ(18) 3 … view at source ↗
Figure 4
Figure 4. Figure 4: 4.5. Simulation results. The simulation study was carried out using the optim() function in R with the BFGS algorithm. The optimization was initialized at the true parameter value. We [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: The path diagram of Model 3 at time t. conducted 10000 simulations with T = 1, D = 10, and ρ = 0.4. The sample sizes were set to n = 5 × 104 , 105 , 5 × 105 , 106 . Tables 1 and 2 show the number of times each model was selected by QAIC and QBIC. Whereas QBIC tends to select the optimal model (Model 1) as the sample size increases, QAIC often selects the overfitted model (Model 2) even when the sample size… view at source ↗
read the original abstract

Structural equation modeling (SEM) is a statistical method for analyzing relationships among latent variables. Since SEM is a confirmatory method, the model needs to be specified in advance. In practice, however, statisticians have several candidate models and aim to select the most appropriate one among them. In this paper, we consider model selection in SEM for jump-diffusion processes. We propose a quasi-Bayesian information criterion (QBIC) for the SEM and show that the proposed criterion has model-selection consistency.

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 / 1 minor

Summary. The paper proposes a quasi-Bayesian information criterion (QBIC) for structural equation modeling (SEM) of jump-diffusion processes observed at high frequency and asserts that the criterion is consistent for selecting among candidate SEMs.

Significance. If the consistency theorem is established under explicitly stated regularity conditions on the underlying jump-diffusion (drift, diffusion coefficient, jump measure) and the SEM parameterization (identifiability, local asymptotic normality of the quasi-likelihood), the result would supply a practical model-selection tool that extends BIC-type criteria to latent-variable stochastic-process settings common in financial econometrics.

major comments (1)
  1. [Abstract] Abstract: the central claim of model-selection consistency is asserted without any indication of the required assumptions, the explicit form of the quasi-likelihood, or even a proof sketch. Because the consistency result is the load-bearing contribution, the manuscript must supply a theorem stating the precise conditions under which the QBIC selects the true SEM with probability tending to 1.
minor comments (1)
  1. The title phrasing 'QBIC of SEM' is slightly awkward; 'QBIC for SEM' would read more naturally.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the abstract. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of model-selection consistency is asserted without any indication of the required assumptions, the explicit form of the quasi-likelihood, or even a proof sketch. Because the consistency result is the load-bearing contribution, the manuscript must supply a theorem stating the precise conditions under which the QBIC selects the true SEM with probability tending to 1.

    Authors: We agree that the abstract should better signal the content of the consistency result. The body of the manuscript (Section 3) contains a theorem that establishes model-selection consistency of the QBIC under explicitly stated regularity conditions on the underlying jump-diffusion process (bounded drift and diffusion coefficients, finite activity jumps with integrable compensator) and on the SEM parameterization (identifiability of the loading matrix and local asymptotic normality of the quasi-likelihood). The quasi-likelihood is defined explicitly in Section 2 as the product of the Gaussian transition densities for the continuous part (via Euler discretization) and the Poisson-like jump component. A complete proof appears in the appendix. We will revise the abstract to include a concise statement of the theorem together with the main regularity conditions and will add a one-sentence pointer to the quasi-likelihood definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proposal plus consistency theorem is self-contained

full rationale

The paper proposes a QBIC for SEM on jump-diffusion processes from high-frequency data and proves model-selection consistency under explicitly stated regularity and identifiability conditions on the drift, diffusion, jump measure, and latent-variable structure. No step reduces by construction to a fitted parameter renamed as a prediction, a self-citation chain, or an ansatz smuggled in from prior work by the same authors. The central claim rests on standard quasi-likelihood asymptotics (local asymptotic normality, uniform convergence) applied to the SEM parameterization; these are independent of the target consistency result and are not shown to be equivalent to the inputs by the paper's own equations. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone provides no identifiable free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5368 in / 829 out tokens · 43157 ms · 2026-05-10T14:56:50.013068+00:00 · methodology

discussion (0)

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Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    and Gloter, A

    Amorino, C. and Gloter, A. (2021). Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function.Statistical Inference for Stochastic Processes,24, 61-148. QBIC OF SEM FOR JUMP-DIFFUSION PROCESSES 41

    work page 2021

  2. [2]

    Czir´ aky, D. (2004). Estimation of dynamic structural equation models with latent variables.Advances in Methodology and Statistics,1(1), 185-204

    work page 2004

  3. [3]

    and Masuda, H

    Eguchi, S. and Masuda, H. (2018). Schwarz type model comparison for LAQ models.Bernoulli,24(3), 2278-2327

    work page 2018

  4. [4]

    and Masuda, H

    Eguchi, S. and Masuda, H. (2024). Gaussian quasi-information criteria for ergodic L´ evy driven SDE. Annals of the Institute of Statistical Mathematics,76(1), 111-157

    work page 2024

  5. [5]

    and Masuda, H

    Eguchi, S. and Masuda, H. (2026). Robustified Gaussian quasi-BIC for volatility. arXiv preprint arXiv:2603.29463

    work page arXiv 2026

  6. [6]

    and Uehara, Y

    Eguchi, S. and Uehara, Y. (2021). Schwartz-type model selection for ergodic stochastic differential equation models.Scandinavian Journal of Statistics,48(3), 950-968

    work page 2021

  7. [7]

    (1984)An introduction to latent variable models,Springer Science & Business Media

    Everitt, B. (1984)An introduction to latent variable models,Springer Science & Business Media

    work page 1984

  8. [8]

    and Uchida, M

    Fujii, T. and Uchida, M. (2014). AIC type statistics for discretely observed ergodic diffusion processes. Statistical Inference for Stochastic Processes,17, 267-282

    work page 2014

  9. [9]

    and Jacod, J

    Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multidimensional diffusion processes.Annales de l’Institut Henri Poincar´ e (B) Probabilit´ es et Statistiques,29, 119-151

    work page 1993

  10. [10]

    Huang, P. H. (2017). Asymptotics of AIC, BIC, and RMSEA for model selection in structural equation modeling.Psychometrika,82(2), 407-426

    work page 2017

  11. [11]

    and Yoshida, N

    Inatsugu, H. and Yoshida, N. (2021). Global jump filters and quasi-likelihood analysis for volatility. Annals of the Institute of Statistical Mathematics,73(3), 555-598

    work page 2021

  12. [12]

    Jasra, A., Kamatani, K., and Masuda, H. (2019). Bayesian inference for stable L´ evy-driven stochastic differential equations with high-frequency data.Scandinavian Journal of Statistics,46(2), 545-574

    work page 2019

  13. [13]

    Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations.Scandinavian Journal of Statistics,24(2), 211-229

    work page 1997

  14. [14]

    and Uchida, M

    Kusano, S. and Uchida, M. (2024). Sparse inference of structural equation modeling with latent variables for diffusion processes.Japanese Journal of Statistics and Data Science,7, 101-150

    work page 2024

  15. [15]

    and Uchida, M

    Kusano, S. and Uchida, M. (2025a). Quasi-Akaike information criterion of SEM with latent variables for diffusion processes.Japanese Journal of Statistics and Data Science,8(1), 217-264

    work page

  16. [16]

    and Uchida, M

    Kusano, S. and Uchida, M. (2025b). Quasi-Bayesian information criterion of SEM for diffusion processes based on high-frequency data.Statistical Inference for Stochastic Processes,28(2), 9

    work page

  17. [17]

    and Uchida, M

    Kusano, S. and Uchida, M. (2025c). Statistical inference in SEM for diffusion processes with jumps based on high-frequency data.Statistical Inference for Stochastic Processes,28(3), 23

    work page

  18. [18]

    and Uchida, M

    Kusano, S. and Uchida, M. (2025d). Akaike-type information criterion of SEM for jump-diffusion processes based on high-frequency data. arXiv preprint arXiv:2511.14333

    work page arXiv

  19. [19]

    Mancini, C. (2004). Estimation of the characteristics of the jumps of a general poisson-diffusion model. Scandinavian Actuarial Journal,2004(1), 42–52

    work page 2004

  20. [20]

    Mueller, R. O. (1999).Basic principles of structural equation modeling: An introduction to LISREL and EQS. Springer Science & Business Media

    work page 1999

  21. [21]

    and Yoshida, N

    Ogihara, T. and Yoshida, N. (2011). Quasi-likelihood analysis for the stochastic differential equation with jumps.Statistical inference for stochastic processes,14, 189-229. 42 S. KUSANO AND M. UCHIDA

    work page 2011

  22. [22]

    Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures.South African Statistical Journal,17(1), 33-81

    work page 1983

  23. [23]

    Shapiro, A. (1985). Asymptotic equivalence of minimum discrepancy function estimators to GLE estimators. SouthAfrican Statistical Journal,19(1), 73-81

    work page 1985

  24. [24]

    and Yoshida, N

    Shimizu, Y. and Yoshida, N. (2003). Janpugata kakusankatei no risankansokukarano suiteinitsuite. (Estimation of diffusion processes with jumps based on discretely sampled observations). The 2003 Japanese Joint Statistical Meeting, Meijo University

    work page 2003

  25. [25]

    and Yoshida, N

    Shimizu, Y. and Yoshida, N. (2006). Estimation of parameters for diffusion processes with jumps from discrete observations.Statistical Inference for Stochastic Processes,9, 227-277

    work page 2006

  26. [26]

    Uchida, M. (2010). Contrast-based information criterion for ergodic diffusion processes from discrete observations.Annals of the Institute of Statistical Mathematics,62, 161-187

    work page 2010

  27. [27]

    and Yoshida, N

    Uchida, M. and Yoshida, N. (2012). Adaptive estimation of an ergodic diffusion process based on sampled data.Stochastic Processes and their Applications,122(8), 2885-2924

    work page 2012

  28. [28]

    Uehara, Y. (2025). Predictive information criterion for jump diffusion processes. arXiv preprint arXiv:2508.00411

    work page arXiv 2025

  29. [29]

    Yoshida, N. (1992). Estimation for diffusion processes from discrete observation.Journal of Multivariate Analysis,41, 220–242

    work page 1992