Bias in the Estimation of the Mean Reversion Parameter in Continuous Time Models

It is well known that for continuous time models with a linear drift standard estimation methods yield biased estimators for the mean reversion parameter both in finite discrete samples and in large in-fill samples. In this paper, we derive two expressions to approximate the bias of the least squares/maximum likelihood estimator of the mean reversion parameter in the Ornstein-Uhlenbeck process with a known long run mean and with an unknown long run mean. The first expression mimics the bias formula of Marriott and Pope (1954) for the discrete time model and corresponds to the bias formula of Tang and Chen (2007). Simulations show that this expression does not work satisfactorily when the speed of mean reversion is slow. Slow mean reversion corresponds to the near unit root situation and is empirically realistic for many financial time series. An improvement is made in the second expression where a nonlinear correction term is included into the bias formula. It is shown that the nonlinear term is important in the near unit root situation. Simulations indicate that the second expression captures the magnitude, the curvature and the non-monotonicity of the actual bias better than the first expression.