The parameter inference for nearly nonstationary time series

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Original languageEnglish
Pages (from-to)857-862
Journal / PublicationJournal of the American Statistical Association
Issue number403
Publication statusPublished - Sept 1988
Externally publishedYes


A first-order autoregressive (AR) time series Y<sub>t</sub>= βY<sub>t−1</sub>+ ε<sub>t</sub>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. © 1976 Taylor & Francis Group, LLC.

Research Area(s)

  • Eigenvalue–eigenfunction expansion, Least Squares, Ornstein-Uhlenbeck process, Percentiles, Stochastic integral

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