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Explosive AR(1) process with independent but not identically distributed errors

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Abstract

Anderson (The Annals of Mathematical Statistics 30(3):676–687, 1959) studied the limiting distribution of the least square estimator for explosive AR(1) process under the independent and identically distributed (iid) condition on error i.e., \(X_t=\rho X_{t-1}+e_t\) where \(\rho >1\) and \(e_t\) is iid error with \(Ee=0\) and \(Ee^2<\infty \). This paper is mainly concerned about the limiting distribution of the least square estimator of \(\rho \), that is \(\hat{\rho }\), when errors are not identically distributed. In addition, we provide an approximate description of the limiting distribution of \(\sum _{j=0}^{n-1}\rho ^{-j}e_{n-j}\) when \(\rho >1\) as \(n\rightarrow \infty \).

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Acknowledgements

We would like to thank the Editor, an AE, and the two referees for their careful reading and valuable comments and helpful suggestions. This work was supported by a grant from the National Research Foundation of Korea (NRF-2019R1F1A1060152(TY Kim), NRF-2018R1A2B2004157(SY Hwang)).

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Authors and Affiliations

  1. Department of Statistics, Keimyung University, Daegu, Korea

    Tae Yoon Kim

  2. Department of Statistics, Sookmyung Womens University, Seoul, Korea

    Sun Young Hwang

  3. Department of Information and Statistics and Research Institute of Natural Science, Gyeongsang National University, Jinju, Korea

    Haejune Oh

Authors
  1. Tae Yoon Kim
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  2. Sun Young Hwang
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  3. Haejune Oh
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Correspondence to Haejune Oh.

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Kim, T.Y., Hwang, S.Y. & Oh, H. Explosive AR(1) process with independent but not identically distributed errors. J. Korean Stat. Soc. 49, 702–721 (2020). https://doi.org/10.1007/s42952-019-00032-w

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  • DOI: https://doi.org/10.1007/s42952-019-00032-w

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