The Parameter Inference for Nearly Nonstationary Time Series

Abstract A first-order autoregressive (AR) time series Yt = βY t-1 + εt is said to be nearly nonstationary if β is close to 1. For a nearly nonstationary AR(1) model, it is shown that the limiting distribution of the least squares estimate of β obtained by Chan and Wei (1987) can be expressed as a functional of the Ornstein-Uhlenbeck process. This alternative expression provides a simple and efficient means to tabulate the percentiles of the limiting distribution that furnishes a useful procedure to test for near nonstationarity. Based on the eigenvalue-eigenfunction consideration, it is shown that the Ornstein-Uhlenbeck formulation possesses an infinite series expansion that extends the result of Dickey and Fuller (1979) to the nearly nonstationary model. Numerical calculations based on different representations of the limiting distribution are performed and compared. It is found that the Ornstein-Uhlenbeck expression provides a better algorithm for tabulating the percentiles. Applications to other time series are also considered. Key Words: Eigenvalue-eigenfunction expansionLeast SquaresOrnstein-Uhlenbeck processPercentilesStochastic integral