A possibly nonstationary autoregressive process, of unknown finite order, with possibly infinite-variance innovations is studied. The ordinary least squares autoregressive parameter estimates are shown to be consistent, and their rate of convergence, which depends on the index of stability, alpha, is established. We also establish consistency of lag-order selection criteria in the nonstationary case. A small experiment illustrates the relative performance of different lag-length selection criteria in finite samples. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
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