Estimation in Two Classes of Semiparametric Diffusion Models

In this paper we propose an estimation method for two classes of semiparametric scalar diffusion models driven by a Brownian motion: In the first class, only the diffusion term is parameterised while the drift is unspecified; in the second, the drift term is specified while the diffusion term is of unknown form. The estimation method is based on the assumption of stationarity of the observed process. This allows us to express the unspecified term as a functional of the parametric part and the stationary density. A MLE-like estimator for the parametric part and a kernel estimator of the nonparametric part are defined for a discrete sample with a fixed time distance between the observations. We show that the parametric part of the estimator is n-consistent, while the nonparametric part has a slower convergence rate. Also, the asymptotic distribution of the estimator is derived. We give a brief discussion of the issue of semiparametric efficiency, and present a small simulation study of the finite-sample performance of our estimator.