  We consider maximum likelihood estimation for both causal and noncausal
autoregressive time series processes with non-Gaussian $\alpha$-stable noise. A
nondegenerate limiting distribution is given for maximum likelihood estimators
of the parameters of the autoregressive model equation and the parameters of
the stable noise distribution. The estimators for the autoregressive parameters
are $n^{1/\alpha}$-consistent and converge in distribution to the maximizer of
a random function. The form of this limiting distribution is intractable, but
the shape of the distribution for these estimators can be examined using the
bootstrap procedure. The bootstrap is asymptotically valid under general
conditions. The estimators for the parameters of the stable noise distribution
have the traditional $n^{1/2}$ rate of convergence and are asymptotically
normal. The behavior of the estimators for finite samples is studied via
simulation, and we use maximum likelihood estimation to fit a noncausal
autoregressive model to the natural logarithms of volumes of Wal-Mart stock
traded daily on the New York Stock Exchange.
