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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