Parameter Estimation for ARMA Models with Infinite Variance Innovations

We consider a standard ARMA process of the form phiBXtThetaBZt, where the innovations Zt belong to the domain of attraction of a stable law, so that neither the Zt nor the Xt have a finite variance. Our aim is to estimate the coefficients of phi and theta. Since maximum likelihood estimation is not a viable possibility due to the unknown form of the marginal density of the innovation sequence we adopt the so-called Whittle estimator, based on the sample periodogram of the X sequence. Despite the fact that the periodogram does not, a priori, seem like a logical object to study in this non-L situation, we show that our estimators are consistent, obtain their asymptotic distributions, and show that they converge to the true values faster than in the usual L2 case.