Inference for near-integrated time series with infinite variance

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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)1069-1074
Journal / PublicationJournal of the American Statistical Association
Volume85
Issue number412
Publication statusPublished - Dec 1990
Externally publishedYes

Abstract

An autoregressive time series is said to be near-integrated (nearly nonstationary) if some of its characteristic roots are close to the unit circle. Statistical inference for the least squares estimators of near-integrated AR(1) models has been under rigorous study recently both in the statistics and econometric literatures. Although classical asymptotics are no longer available, through the study of weak convergence of stochastic processes, one can establish the asymptotic theories in terms of simple diffusion processes or Brownian motions. Such results rely heavily on the finiteness of the variance of the noise. When this finite variance condition fails, whereas many physical and economic phenomena are believed to be generated by an infinite variance noise sequence, the aforementioned asymptotics are not applicable. In this article, a unified theory concerning near-integrated autoregressive time series with infinite variance is developed. In particular, when the noise sequence {ε<sub>t</sub>} belongs to the domain of attraction of a stable law with index α (0 &lt; α &lt; 2), that is, heavy-tailed, the asymptotic distribution of the least squares estimate of the autoregressive coefficient of a near-integrated AR(1) model is established. Instead of Brownian motions, weak convergence theory and stochastic integral involving Lévy processes are required to handle the infinite variance case. Results developed in this article not only provide an analog parallel to the finite variance model, but also allow one to study issues involving unit root tests and cointegrations where the underlying time series is generated by a heavy-tailed noise sequence. © 1990 Taylor & Francis Group, LLC.

Research Area(s)

  • Heavy-tailed noise, Least squares, Lévy process, Stable distribution, Stochastic integral

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