Pseudo-Maximum Likelihood Estimation in Two Classes of Semiparametric Diffusion Models

A novel estimation method for two classes of semiparametric scalar diffusion models is proposed: In the first class, the diffusion term is parameterised and the drift is left unspecified, while in the second class only the drift term is specified. Under the assumption of stationarity, the unspecified term can be identified as a functional of the parametric component and the stationary density. Given a discrete sample with a fixed time distance, the parametric component is then estimated by maximizing the associated likelihood with a preliminary estimator of the unspecified term plugged in. It is shown that this Pseudo-MLE (PMLE) is root n-consistent and asymptotically normally distributed under regularity conditions, and demonstrate how the models and estimators can be used in a two-step specification testing strategy of fully parametric models. Since the likelihood function is not available on closed form, the practical implementation of our estimator and tests will rely on simulated or approximate PMLE's. Under regularity conditions, it is verified that approximate/simulated versions of the PMLE inherits the properties of the actual but infeasible estimator. A simulation study investigates the finite-sample performance of the PMLE, and finds that it performs well and is comparable to parametric MLE both in terms of bias and variance.