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

TL;DR

This paper investigates boundary inference at the critical H=3/4 for mixed fractional models under high-frequency data, establishing CLTs, LAN, and boundary tests with practical applications.

Contribution

It identifies the critical scaling, derives explicit LAN and score tests at H=3/4, and develops boundary-calibrated tests for supercritical detection.

Findings

Critical score CLT and LAN established at H=3/4

Boundary-calibrated one-sided score tests constructed

Empirical analysis shows no evidence of H>3/4 in SPY data

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 $(\sigma,H)$ 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 $H>3/4$.