Bootstrapping Unstable First-Order Autoregressive Processes

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