Prediction-based estimation for diffusion models with high-frequency data

TL;DR

This paper develops asymptotic theory for prediction-based estimators in high-frequency diffusion models, proving consistency and normality under specific conditions, and introduces a Monte Carlo method for variance estimation.

Contribution

It provides new asymptotic results for prediction-based estimators in high-frequency diffusion data, including consistency, normality, and a Monte Carlo variance estimation method.

Findings

Proved consistency of prediction-based estimators.

Established asymptotic normality under rate conditions.

Proposed Monte Carlo method for variance calculation.

Abstract

This paper obtains asymptotic results for parametric inference using prediction-based estimating functions when the data are high frequency observations of a diffusion process with an infinite time horizon. Specifically, the data are observations of a diffusion process at $n$ equidistant time points $\Delta_n i$, and the asymptotic scenario is $\Delta_n \to 0$ and $n\Delta_n \to \infty$. For a useful and tractable classes of prediction-based estimating functions, existence of a consistent estimator is proved under standard weak regularity conditions on the diffusion process and the estimating function. Asymptotic normality of the estimator is established under the additional rate condition $n\Delta_n^3 \to 0$. The prediction-based estimating functions are approximate martingale estimating functions to a smaller order than what has previously been studied, and new non-standard asymptotic theory is needed. A Monte Carlo method for calculating the asymptotic variance of the estimators is proposed.