In-fill asymptotic theory for structural break point in autoregressions

This article obtains the exact distribution of the maximum likelihood estimator of structural break point in the Ornstein–Uhlenbeck process when a continuous record is available. The exact distribution is asymmetric, tri-modal, dependent on the initial condition. These three properties are also found in the finite sample distribution of the least squares (LS) estimator of structural break point in autoregressive (AR) models. Motivated by these observations, the article then develops an in-fill asymptotic theory for the LS estimator of structural break point in the AR(1) coefficient. The in-fill asymptotic distribution is also asymmetric, tri-modal, dependent on the initial condition, and delivers excellent approximations to the finite sample distribution. Unlike the long-span asymptotic theory, which depends on the underlying AR roots and hence is tailor-made but is only available in a rather limited number of cases, the in-fill asymptotic theory is continuous in the underlying roots. Monte Carlo studies show that the in-fill asymptotic theory performs better than the long-span asymptotic theory for cases where the long-span theory is available and performs very well for cases where no long-span theory is available. The article also proposes to use the highest density region to construct confidence intervals for structural break point.