Parameter Estimation for ARMA Models with Infinite Variance Innovations

We consider a standard ARMA process of the form φ(B)Xₜ = θ(B)Zₜ, 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 φ and θ. 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-L² 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.