Parameter Estimation for ARMA Models with Infinite Variance Innovations

摘要

We consider a standard ARMA process of the form Phi(B)X sub t = Theta(B)Z sub t,where the special innovations Z sub t belong to the domain of attraction of a stable law, so that neither the Z sub t nor the X sub t 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-sq 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 L-sq case.