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Akaike-type information criterion of SEM for jump-diffusion processes based on high-frequency data

Shogo Kusano, Masayuki Uchida

TL;DR

This work develops an Akaike-type information criterion (QAIC) for structural equation modeling (SEM) when the underlying processes are jump-diffusions observed at high frequency. The authors construct a locally Gaussian quasi-likelihood with a jump-detection rule and define QAIC_n(m) = -2H_{m,n}(\mathbb{X}_n,\hat{\theta}_{m,n}) + 2q_m to enable model selection among SEM candidates with latent factors. They prove moment convergence and asymptotic normality of the quasi-maximum likelihood estimators, establish asymptotic unbiasedness of the QAIC, and analyze model selection in both correctly specified and misspecified settings. A simulation study demonstrates consistency of estimators and shows QAIC reliably excludes the true misspecified model while tending to over-select over-fitted models in large samples, highlighting both the method’s strengths and limitations in finite-sample SEM with jumps. The results provide a principled tool for SEM selection in continuous-time jump-diffusion contexts relevant to finance and other domains with high-frequency data.

Abstract

Structural equation modeling (SEM) is a statistical method used to investigate relationships among latent variables. In SEM, the model must be specified in advance. However, in practice, statisticians often have several candidate models and need to select the most appropriate one. Consequently, model selection is a key issue in SEM, and information criteria are commonly used to address this issue. In this study, we develop an Akaike-type information criterion of SEM for jump-diffusion processes, which enables model selection for SEM based on high-frequency data with jumps. Simulation studies are conducted to illustrate the finite-sample performance of the proposed method.

Akaike-type information criterion of SEM for jump-diffusion processes based on high-frequency data

TL;DR

This work develops an Akaike-type information criterion (QAIC) for structural equation modeling (SEM) when the underlying processes are jump-diffusions observed at high frequency. The authors construct a locally Gaussian quasi-likelihood with a jump-detection rule and define QAIC_n(m) = -2H_{m,n}(\mathbb{X}_n,\hat{\theta}_{m,n}) + 2q_m to enable model selection among SEM candidates with latent factors. They prove moment convergence and asymptotic normality of the quasi-maximum likelihood estimators, establish asymptotic unbiasedness of the QAIC, and analyze model selection in both correctly specified and misspecified settings. A simulation study demonstrates consistency of estimators and shows QAIC reliably excludes the true misspecified model while tending to over-select over-fitted models in large samples, highlighting both the method’s strengths and limitations in finite-sample SEM with jumps. The results provide a principled tool for SEM selection in continuous-time jump-diffusion contexts relevant to finance and other domains with high-frequency data.

Abstract

Structural equation modeling (SEM) is a statistical method used to investigate relationships among latent variables. In SEM, the model must be specified in advance. However, in practice, statisticians often have several candidate models and need to select the most appropriate one. Consequently, model selection is a key issue in SEM, and information criteria are commonly used to address this issue. In this study, we develop an Akaike-type information criterion of SEM for jump-diffusion processes, which enables model selection for SEM based on high-frequency data with jumps. Simulation studies are conducted to illustrate the finite-sample performance of the proposed method.

Paper Structure

This paper contains 13 sections, 16 theorems, 312 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and Assumptions
  3. Main results
  4. Examples and Simulation results
  5. True model
  6. Model 1
  7. Model 2
  8. Model 3
  9. Simulation results
  10. Proofs
  11. Appendix
  12. Proof of Lemma \ref{['EX2abslemma']}
  13. The check of identifiability (\ref{['model2iden']})

Key Result

Theorem 1

Suppose that [A1]-[A4] and [B1]. Then, for all $L>0$, there exists $C_L>0$ such that for all $r>0$ and $n\in\mathbb{N}$.

Figures (4)

  • Figure 1: The path diagram of the true model at time $t$.
  • Figure 2: The path diagram of Model 1 at time $t$.
  • Figure 3: The path diagram of Model 2 at time $t$.
  • Figure 4: The path diagram of Model 3 at time $t$.

Theorems & Definitions (30)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Proposition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 20 more