Modified least squares estimators for Ornstein–Uhlenbeck processes from low-frequency observations

Abstract

We propose a modified least squares estimator for the drift parameter of the Ornstein–Uhlenbeck process when the observations are available at a discrete instant in a low-frequency level. Unlike in the past literature, this modified least squares estimator is asymptotically unbiased. This estimator is combined with the ergodic theorem to obtain joint estimators (θˆn,σˆn2) for both drift and diffusion parameters (θ,σ2). For any fixed observation time step h>0, the strong consistency and joint asymptotic normality of our estimators (θˆn,σˆn2) are obtained by using the linear model technology and the Delta method. Surprisingly, this linear model technology is not new but it is used for the first time to our model. It very significantly simplifies the arguments that researchers used in the literature. Numerical results illustrate the asymptotic behavior of the estimators.