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

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 \(\Delta _n \rightarrow 0\) and \(n\Delta _n \rightarrow \infty\). For 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 \(n\Delta _n^3 \rightarrow 0\). 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.