Exact Distribution of the Mean Reversion Estimator in the Ornstein-Uhlenbeck Process

Econometricians have recently been interested in estimating and testing the mean reversion parameter (κ) in linear diffusion models. It has been documented that the maximum likelihood estimator (MLE) of κ tends to over estimate the true value. Its asymptotic distribution, on the other hand, depends on how the data are sampled (under expanding, infill, or mixed domain) as well as how we spell out the initial condition. This poses a tremendous challenge to practitioners in terms of estimation and inference. In this paper, we provide new and significant results regarding the exact distribution of the MLE of κ in the Ornstein-Uhlenbeck process under different scenarios: known or unknown drift term, fixed or random start-up value, and zero or positive κ. In particular, we employ numerical integration via analytical evaluation of a joint characteristic function. Our numerical calculations demonstrate the remarkably reliable performance of our exact approach. It is found that the true distribution of the MLE can be severely skewed in finite samples and that the asymptotic distributions in general may provide misleading results. Our exact approach indicates clearly the non-mean-reverting behavior of the real federal fund rate.