New distribution theory for the estimation of structural break point in mean

Based on the Girsanov theorem, this paper obtains the exact distribution of the maximum likelihood estimator of structural break point in a continuous time model. The exact distribution is asymmetric and tri-modal, indicating that the estimator is biased. These two properties are also found in the finite sample distribution of the least squares (LS) estimator of structural break point in the discrete time model, suggesting the classical long-span asymptotic theory is inadequate. The paper then builds a continuous time approximation to the discrete time model and develops an in-fill asymptotic theory for the LS estimator. The in-fill asymptotic distribution is asymmetric and tri-modal and delivers good approximations to the finite sample distribution. To reduce the bias in the estimation of both the continuous time and the discrete time models, a simulation-based method based on the indirect estimation (IE) approach is proposed. Monte Carlo studies show that IE achieves substantial bias reductions.