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Bootstrapping unstable first-order autoregressive processes. (English) Zbl 0725.62076

Summary: Consider a first-order autoregressive process \(X_ t=\beta X_{t- 1}+\epsilon_ t\), where \(\{\epsilon_ t\}\) are independent and identically distributed random errors with mean 0 and variance 1. It is shown that when \(\beta =1\) the standard bootstrap least squares estimate of \(\beta\) is asymptotically invalid, even if the error distribution is assumed to be normal. The conditional limit distribution of the bootstrap estimate at \(\beta =1\) is shown to converge to a random distribution.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M09 Non-Markovian processes: estimation
62E20 Asymptotic distribution theory in statistics
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