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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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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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