Zarepour, M.; Knight, K. Bootstrapping unstable first order autoregressive process with errors in the domain of attraction of stable law. (English) Zbl 0919.62110 Commun. Stat., Stochastic Models 15, No. 1, 11-27 (1999). Summary: We consider a first order autoregressive process \(X_t= \varphi X_{t-1} +\varepsilon_t\) where \(\{\varepsilon_t\}\) are independent and identically distributed random errors which belong to the domain of attraction of a stable law with index \(\alpha\in (0,2]\). We show that when \(\varphi =1\), Efron’s bootstrap of the \(M\)-estimator of \(\varphi\) is asymptotically valid if and only if \(E(\varepsilon^2_1) =\infty\). Also we show that for \(0< \alpha<2\) the usual bootstrap is asymptotically invalid for the least-squares estimate. A modification of the resampling sample size to \(m_n= o(n)\) makes the bootstrap asymptotically valid for all \(\alpha\in (0,2]\). Cited in 6 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators 60E07 Infinitely divisible distributions; stable distributions Keywords:random walk; autoregressive process; stable law; bootstrap PDFBibTeX XMLCite \textit{M. Zarepour} and \textit{K. Knight}, Commun. Stat., Stochastic Models 15, No. 1, 11--27 (1999; Zbl 0919.62110) Full Text: DOI