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Boundary Inference for Mixed Fractional Models under High-Frequency Observation Critical LAN and Score Tests at $H=3/4$

Chunhao Cai, Yiwu Shang, Weilin Xiao, Cong Zhang

Abstract

We study boundary inference at $H=3/4$ for mixed fractional Brownian motion and mixed fractional Ornstein--Uhlenbeck models under high-frequency observation. This boundary is economically important because it separates the critical and supercritical regimes of mixed fractional dynamics. We make three contributions. First, we identify the exact critical first-order scaling and show that, after removing the explicit linear component in the $H$-score, the transformed $(σ,H)$ block is already non-degenerate. Second, we establish critical score central limit theorems (CLT) and derive local asymptotic normality (LAN) with fully explicit leading information constants for both models. Third, we construct boundary-calibrated one-sided score tests for detecting entry into the supercritical region $H>3/4$ and discuss feasible implementation through restricted nuisance estimation. Monte Carlo evidence shows that the feasible statistic has the correct directional power but conservative null calibration. Finally, an intraday illustration on one-minute SPY data finds no persistent evidence in favor of $H>3/4$.

Boundary Inference for Mixed Fractional Models under High-Frequency Observation Critical LAN and Score Tests at $H=3/4$

Abstract

We study boundary inference at for mixed fractional Brownian motion and mixed fractional Ornstein--Uhlenbeck models under high-frequency observation. This boundary is economically important because it separates the critical and supercritical regimes of mixed fractional dynamics. We make three contributions. First, we identify the exact critical first-order scaling and show that, after removing the explicit linear component in the -score, the transformed block is already non-degenerate. Second, we establish critical score central limit theorems (CLT) and derive local asymptotic normality (LAN) with fully explicit leading information constants for both models. Third, we construct boundary-calibrated one-sided score tests for detecting entry into the supercritical region and discuss feasible implementation through restricted nuisance estimation. Monte Carlo evidence shows that the feasible statistic has the correct directional power but conservative null calibration. Finally, an intraday illustration on one-minute SPY data finds no persistent evidence in favor of .
Paper Structure (27 sections, 19 theorems, 215 equations, 4 figures, 4 tables)

This paper contains 27 sections, 19 theorems, 215 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model setup, critical objects and score decomposition
  3. The mfBm model
  4. The mfOU model
  5. Critical covariance constants
  6. Exact score functions
  7. Critical profiles.
  8. Main asymptotic results
  9. Critical score central limit theorems
  10. Critical LAN expansions
  11. Technical proof of the score CLTs
  12. Critical trace asymptotics
  13. CLT for the transformed score vector
  14. Technical proof of the LAN theorems
  15. Matrix Taylor expansions
  16. ...and 12 more sections

Key Result

Proposition 1

Let Then, as $\lambda\to0$, and where $\psi$ denotes the digamma function. In particular, at $H=3/4$, and $\beta_{\mathrm m}$ is the derivative of $\log c_H$ at $H=3/4$.

Figures (4)

  • Figure 1: Empirical rejection rates of the feasible mfBm boundary score across the supercritical power grid in Experiment I.
  • Figure 2: Empirical standard deviation of the feasible mfBm boundary score under the critical null across the asymptotic sequence in Experiment II.
  • Figure 3: Daily exact feasible mfBm statistic on one-minute SPY returns, with 60-day rolling mean and one-sided critical lines.
  • Figure 4: 60-day rolling rejection share of the exact feasible mfBm statistic at the nominal $5\%$ level.

Theorems & Definitions (39)

  • Proposition 1
  • proof
  • Corollary 2
  • proof
  • Remark 3
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 29 more