M-ESTIMATION FOR AUTOREGRESSIONS WITH INFINITE VARIANCE

We study the problem of estimating autoregressive parameters when the observations are from an AR process with innovations in the domain of attraction of a stable law. We show that non-degenerate limit laws exist for M-estimates if the loss function is sufficiently smooth; these results remain valid if location and scale are also estimated. For least absolute deviation (LAD) estimates, similar results hold under conditions on the innovations distribution near 0. We also discuss, under moment conditions on the innovations, consistency properties for M-estimators corresponding to the class of loss functions, rho(x) = \x\gamma for some gamma > 0.