## Maximum likelihood estimation for $$\alpha$$-stable autoregressive processes.(English)Zbl 1168.62077

Summary: We consider maximum likelihood estimation for both causal and non-causal 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 a 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 simulations, and we use maximum likelihood estimation to fit a noncausal autoregressive model to the natural logarithms of the volumes of Wal-Mart stock traded daily on the New York Stock Exchange.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E20 Asymptotic distribution theory in statistics 62F40 Bootstrap, jackknife and other resampling methods 62F12 Asymptotic properties of parametric estimators 62F10 Point estimation

### Software:

STABLE; fminsearch; SYMSTB
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