Accelerated Asymptotics for Diffusion Model Estimation

We propose a semiparametric estimation procedure for scalar homogeneous stochastic differential equations. We specify a parametric class for the underlying diffusion process and identify the parameters of interest by minimizing criteria given by the integrated squared difference between kernel estimates of drift and diffusion function and their parametric counterparts. The nonparametric estimates are simplified versions of those in Bandi and Phillips (1998). A complete asymptotic theory for the semiparametric estimates is developed. The limit theory relies on infill and long span asymptotics and the asymptotic distributions are shown to depend on the chronological local time of the underlying diffusion process. The estimation method and asymptotic results apply to both stationary and nonstationary processes. As is standard with semiparametric approaches in other contexts, faster convergence rates are attained than is possible in the fully functional case. From a purely technical point of view, this work merges two strands of the most recent econometrics literature, namely the estimation of nonlinear models of integrated time-series [Park and Phillips (1999, 2000)] and the functional identification of diffusions under minimal assumptions on the dynamics of the underlying process [Florens-Zmirou (1993), Jacod (1997), Bandi and Phillips (1998) and Bandi (1999)]. In effect, the 'minimum distance' type of estimation that is presented in this paper can be interpreted as extremum estimation for potentially nonstationary and nonlinear continuous-time models.