Statistical inference in SEM for diffusion processes with jumps based on high-frequency data

TL;DR

This work develops SEM for diffusion processes with jumps observed at high frequency, introducing a threshold-based quasi-likelihood to separate diffusion and jump components. It proves that the threshold estimator of the diffusion covariance ${\\boldsymbol{\\Sigma}}$ and the QMLE for SEM parameters are consistent and asymptotically normal, and that a quasi-likelihood ratio test yields an asymptotic $\\chi^2$ distribution under the null. The approach is validated via extensive simulations, including true, correctly specified, and misspecified models, demonstrating both finite-sample performance and the test’s ability to detect misspecification. The results provide a principled framework for causal and structural inference in SEMs when data exhibit jumps, with direct implications for high-frequency financial, hydrological, and other jump-affected systems.

Abstract

We study structural equation modeling (SEM) for diffusion processes with jumps. Based on high-frequency data, we consider the parameter estimation and the goodness-of-fit test in the SEM. Using a threshold method, we propose the quasi-likelihood of the SEM and prove that the quasi-maximum likelihood estimator has consistency and asymptotic normality. To examine whether a specified parametric model is correct or not, we also construct the quasi-likelihood ratio test statistics and investigate the asymptotic properties. Furthermore, numerical simulations are conducted.

Figures (7)

• Figure 1: Sample path from a diffusion process with jumps.
• Figure 2: Path diagram of the true model at time $t$.
• Figure 3: Path diagram of the correctly specified model at time $t$.
• Figure 4: Path diagram of the misspecified model at time $t$.
• Figure 5: Histogram (left), Q-Q plot (middle) and empirical distribution (right) of $\sqrt{n}((\hat{\bf{\Sigma}}_{n})_{11}-({\bf{\Sigma}}_0)_{11})$. The red lines are theoretical curves.

Theorems & Definitions (37)

• Remark 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof : Proof of Lemma \ref{['Pine']}
• Lemma 4