On the First-Order Autoregressive Process with Infinite Variance

For a first-order autoregressive process <italic>Y</italic> = β<italic>Y</italic>_t−1 + <italic>null</italic> where the null<italic>null</italic>'S are i.i.d. and belong to the domain of attraction of a stable law, the strong consistency of the ordinary least-squares estimator <italic>b</italic> of <italic>β</italic> is obtained for <italic>β</italic> = 1, and the limiting distribution of <italic>b</italic> is established as a functional of a Lévy process. Generalizations to seasonal difference models are also considered. These results are useful in testing for the presence of unit roots when the null<italic>null</italic>'S are heavy-tailed.